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VI International Meetingon Lorentzian Geometry
Facultad de CienciasUniversidad de Granada
Granada, 6–9 September, 2011
Foreword
The Organizing Committee of the VI International Meeting on Lorentzian Ge-ometry, Granada 2011 (gelogra’11), http://gigda.ugr.es/gelogra/, wouldlike to welcome all participants of the meeting. It will be held on September6–9, 2011 at the Science Faculty of the University of Granada, in the beautifuland monumental city of Granada, located in the south of Spain.
Ten years ago and in a friendly atmosphere, several Spanish research groupsinterested on the area of Lorentzian Geometry, its applications and relatedmathematical topics, initiated the biennial Meetings on Lorentzian Geome-try and its Applications in Benalmádena-2001 (Málaga). Since that time, thisseries of meetings has been consolidated, has become international, and for-tunately, now we are in the sixth edition of the event.
Lorentzian Geometry was born as the mathematical tool used in GeneralRelativity. Nowadays, it has own identity and constitutes a very active branchof Differential Geometry where many mathematical techniques are involved(Geometric Analysis, Functional Analysis, Partial Differential Equations, Liegroups and Lie algebras, . . . ).
In this meeting, topics on pure and applied Lorentzian geometry such asgeodesics, submanifolds, causality, black holes, Einstein equations, geome-try of spacetimes or AdS-CFT correspondence, will be covered. The ScientificCommittee has designed an intensive and interesting program of invited lec-tures and mini-courses, as well as a selection of short talks. In addition, awide number of posters will complete the scientific program.
There will also be some cultural and social activities, including a reception atthe Hospital Real, which is the main administrative building of the Universityof Granada, and a enjoyable guided visit to the Alhambra Monument, forthose interested participants.
Finally, we would like to wish everybody who is attending gelogra’11 afruitful stay, both geometrically and personally. This has been our main aimin the organization of this meeting.
Miguel OrtegaFrancisco J. PalomoAlfonso RomeroMiguel SánchezFrancisco Torralbo
gelogra’11, Orgainizing Committee
iv
Committees
Organizing Committee
Miguel OrtegaUniversidad de Granada
Francisco J. PalomoUniversidad de Málaga
Miguel SánchezUniversidad de Granada
Francisco TorralboUniversidad de Granada
Alfonso RomeroUniversidad de Granada
Scientific Committee
Dimitri AlekseevskyUniversity of Edinburgh
Manuel BarrosUniversidad de Granada
Anna Maria CandelaUniversità degli studi di Bari
Yvonne Choquet-BruhatIHES France
Paul EhrilchUniversity of Florida
Ángel FerrándezUniversidad de Murcia
José Luis FloresUniversidad de Málaga
Eduardo García-RíoUniversidad de Santiagode Compostela
Hans-Bert RademacherUniversität Leipzig
Collaborator Entities
Research projects
Research Group Differential Geometry and its applications FQM-324
http://gigda.ugr.es/gigda/
Research project Lorentz Geometry and Gravitation P09-FQM-4496
http://gigda.ugr.es/glg/
Research project Semi-Riemannian Geometry and Variational Problems inMathematical Physics MTM2010-18099
http://gigda.ugr.es/pm2011/
Institutions
Facultad de Ciencias
Sponsors
Programm
e
Tuesday, September 6, 2011
8:15–8:45 Registration
8:45–9:00 Opening
9:00–10:00 Invited speakerIndefinite extrinsic symmetric spacesI. Kath, Universität Greifswald . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
10:00–11:00 Invited speakerHow to reconstruct a metric by its unparameterized geodesicsV. Matveev, Friedrich-Schiller-Universität Jena . . . . . . . . . . . . . . . . . . . 17
11:00–11:30 Poster session & Coffee break
11:30–12:00 Short talkOn one class of holonomy groups in pseudo-RiemanniangeometryA. Bolsinov, Loughborough University . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
12:00–12:30 Short talkClasification of symmetric M-theory backgroundsJ.M. Figueroa-O’Farrill, University of Edinburgh . . . . . . . . . . . . . . . . . 37
12:30–12:50 Short talkClosed geodesics in Lorentzian surfacesS. Suhr, Universität Hamburg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
12:50–13:10 Short talkLorentzian quasi-Einstein manifoldsS. Gavino-Fernández, Universidad Santiago de Compostela . . . . . . 31
13:10–13:30 Short talkCharacterization and structure of second-order symmetricLorentzian manifoldsO. F. Blanco, Universidad de Granada . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
13:30–15:30 Lunch
2
15:30–16:00 Short talkThe geometry of collapsing isotropic fluidsR. Giambò, Università degli Studi di Camerino . . . . . . . . . . . . . . . . . . 32
16:00–17:00 Mini courseCausality and Legendrian linking IV. Chernov, Dartmouth College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
17:00–17:30 Poster session & Coffee break
17:30–18:00 Mini courseOn the isometry group of Lorentz manifolds IP. Piccione, Universidade de São Paulo . . . . . . . . . . . . . . . . . . . . . . . . . . 20
18:30–19:00 Short talkOn the geometric structure of Ori spacetimesM. Scherfner, Technische Universität Berlin . . . . . . . . . . . . . . . . . . . . . 37
20:30 Rectorat University Cocktail Reception
Wednesday, September 7, 2011
9:00–10:00 Invited speakerNew examples of maximal surfaces in Lorentz Minkowski3-spaceM. Umehara, K. Yamada, Tokyo Institute of Technology . . . . . . . . . . 18
10:00–11:00 Invited speakerThe c-boundary of spacetimes and its related boundaries indifferential geometryJ. Herrera, Universidad de Málaga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
11:00–11:30 Poster session & Coffee break
11:30–12:00 Short talkSplitting theorems in Lorentzian geometryD. Solis, Universidad Autónoma de Yucatán . . . . . . . . . . . . . . . . . . . . . . . 38
12:00–12:30 Short talkConformally flat homogeneous Lorentzian manifoldsK. Tsukada, Ochanomizu University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3
12:30–12:50 Short talkTotally geodesic submanifolds in Robertson-Walker spaceswith flat fibersZ. Dušek, Palacky University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
12:50–13:10 Short talkSlant geometry of spacelike hypersurfaces in the Lightconewith respect to the φ-hyperbolic dualsH. Yıldırım, Istanbul University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
13:10–13:30 Short talkExtrinsically flat surfaces of space forms and the geometricstructure on the space of oriented geodesicsA. Honda, Tokyo Institute of Technology . . . . . . . . . . . . . . . . . . . . . . . . . 34
13:30–15:30 Lunch
15:30–16:00 Short talkOn the space of geodesics of Riemannian and Lorentzianspace formsH. Anciaux, Universidade de São Paulo . . . . . . . . . . . . . . . . . . . . . . . . . 22
16:00–17:00 Mini courseCausality and Legendrian linkingV. Chernov, Dartmouth College II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
17:00–17:30 Poster session & Coffee break
17:30–18:00 Mini courseOn the isometry group of Lorentz manifoldsP. Piccione, Universidade de São Paulo II . . . . . . . . . . . . . . . . . . . . . . . . 20
18:30–19:00 Short talkGeodesic connectedness on Gödel type spacetimes: a “static”variational set-upR. Bartolo, Politecnico di Bari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
21:00 Social Dinner at “Carmen de los Chapiteles”(See the Social Events section for more information).
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Thursday, September 8, 2011
9:00–10:00 Invited speakerCalabi-Berstein results and parabolicity of maximal surfacesin Lorentzian product spacesL. J. Alías, Universidad de Murcia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
10:00–11:00 Invited speakerPolar actions on symmetric spaces of noncompact typeJ. C. Díaz-Ramos, Universidad de Santiago de Compostela . . . . . . . 14
11:00–11:30 Poster session & Coffee break
11:30–12:00 Short talkKähler and para-Kähler Weyl manifolds of dimension 4P. Gilkey, University of Oregon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
12:00–12:30 Short talkPseudo-umbilical and related surfaces in spacetimeJ. M. M. Senovilla, Universidad del País Vasco . . . . . . . . . . . . . . . . . . . 38
12:30–12:50 Short talkUniqueness and non-existence results for spacelikehypersurfaces with constant mean curvature in Hn ×R1A. Albujer, Universidad de Córdoba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
12:50–13:10 Short talkSpacelike hypersurfaces of constant higher order meancurvature in generalized Robertson-Walker spacetimesD. Impera, Università degli Studi di Milano . . . . . . . . . . . . . . . . . . . . . . 35
13:10–13:30 Short talkFamilies of parallel cmc hypersurfaces in noncompactrank-one symmetric spacesM. Domínguez-Vázquez, Universidad Santiago de Compostela . . 28
13:30–15:30 Lunch
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Friday, September 9, 2011
9:00–10:00 Invited speakerStability of marginally outer trapped surfaces andapplicationsM. Mars, Universidad de Salamanca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
10:00–11:00 Invited speakerStationary-to-Randers correspondence and convexityE. Caponio, Politecnico di Bari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
11:00–11:30 Poster session & Coffee break
11:30–12:00 Short talkArea-angular momentum inequality in stable marginallytrapped surfacesJ. L. Jaramillo, Max-Planck-Institut Gravitationsphysik . . . . . . . . . . . . 36
12:00–12:30 Short talkK-conformal maps of a stationay spacetime and RandersmetricM. A. Javaloyes, Universidad de Murcia . . . . . . . . . . . . . . . . . . . . . . . . . 36
12:30–12:50 Short talkUniqueness of spacelike hypersurfaces of contant meancurvature in cosmological models with certain symmetriesM. Caballero, Universidad de Córdoba . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
12:50–13:10 Short talkPinching the Fermat metric in stationary spacetimesA. Dirmeier, Technische Universität Berlin . . . . . . . . . . . . . . . . . . . . . . . 27
13:10–13:30 Short talkLorentzian Bobillier FormulaS. Ersoy, Sakarya University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
13:30–15:30 Lunch
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16:00–17:00 Mini courseCausality and Legendrian linking IIIV. Chernov, Dartmouth College . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
17:00–17:30 Poster session & Coffee break
17:30–18:00 Mini courseOn the isometry group of Lorentz manifolds IIIP. Piccione, Universidade de São Paulo . . . . . . . . . . . . . . . . . . . . . . . . . . 20
18:30–19:00 Closing Ceremony
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Abstracts
Invited lectures 11
Luis J. Alías . . . . . . . . . . . . 11
Erasmo Caponio . . . . . . . . . . 13
José Carlos Díaz-Ramos . . . . . 14
Jónatan Herrera . . . . . . . . . . 15
Ines Kath . . . . . . . . . . . . . . 16
Marc Mars . . . . . . . . . . . . . 16
Vladimir S. Matveev . . . . . . . 17
Masaaki Umehara and KotaroYamada . . . . . . . . . . . . 18
Minicourses 19
Vladimir Chernov . . . . . . . . . 19
Paolo Piccione . . . . . . . . . . . 20
Short talks 21
Alma L. Albujer . . . . . . . . . . 21
Henri Anciaux . . . . . . . . . . . 22
Rossella Bartolo . . . . . . . . . . 23
Oihane F. Blanco . . . . . . . . . . 23
Alexey Bolsinov . . . . . . . . . . 24
Magdalena Caballero . . . . . . . 26
Alexander Dirmeier . . . . . . . . 27
Miguel Domínguez-Vázquez . . 28
Zdenek Dušek . . . . . . . . . . . 29
Soley Ersoy . . . . . . . . . . . . . 30
S. Gavino-Fernández . . . . . . . 31
Roberto Giambò . . . . . . . . . . 32
P. Gilkey . . . . . . . . . . . . . . 33
Atsufumi Honda . . . . . . . . . 34
Debora Impera . . . . . . . . . . . 35
José Luis Jaramillo . . . . . . . . 36
Miguel A. Javaloyes . . . . . . . . 36
Figueroa O’Farrill . . . . . . . . . 37
Mike Scherfner . . . . . . . . . . 37
José M. M. Senovilla . . . . . . . 38
Didier A. Solis . . . . . . . . . . . 38
Stefan Suhr . . . . . . . . . . . . . 39
Kazumi Tsukada . . . . . . . . . 40
Handan Yıldırım . . . . . . . . . 41
Posters 42
Mahmut Akyigit . . . . . . . . . . 42
Letizia Brunetti . . . . . . . . . . 43
Angelo V. Caldarella . . . . . . . 44
Zdenek Dušek . . . . . . . . . . . 45
Atsufumi Honda . . . . . . . . . 46
Abdullah Inalcik . . . . . . . . . 47
Kosuke Naokawa . . . . . . . . . 48
M. Eugenia Rosado . . . . . . . . 49
Cezar Oniciuc . . . . . . . . . . . 51
H. Fabián Ramírez-Ospina . . . . 51
Ilgin Sager . . . . . . . . . . . . . 52
Marcos Salvai . . . . . . . . . . . 53
J. Seoane-Bascoy . . . . . . . . . . 54
Sergiu I. Vacaru . . . . . . . . . . 56
Invited lectures
Calabi-Bernstein results and parabolicity of maximalsurfaces in Lorentzian product spaces
Luis J. Alías ([email protected])Universidad de Murcia
A maximal surface in a 3-dimensional Lorentzian manifold is a spacelike surfacewith zero mean curvature. Here by spacelike we mean that the induced metric fromthe ambient Lorentzian metric is a Riemannian metric on the surface. The terminol-ogy maximal comes from the fact that these surfaces locally maximize area among allnearby surfaces having the same boundary. Besides their mathematical interest, max-imal surfaces and, more generally, spacelike surfaces with constant mean curvatureare also important in General Relativity.
One of the most important global results about maximal surfaces is the Calabi-Bernstein theorem for maximal surfaces in the 3-dimensional Lorentz-Minkowskispace R3
1, which, in parametric version, states that the only complete maximal sur-faces in R3
1 are the spacelike planes. This result can be seen also in a non-parametricform, establishing that the only entire maximal graphs in R3
1 are the spacelike planes;that is, the only entire solutions to the maximal surface equation
Div
(Du√
1− |Du|2
)= 0, |Du|2 < 1
on the Euclidean plane R2 are affine functions.
In this lecture, we present new Calabi-Bernstein results for maximal surfaces im-mersed into a Lorentzian product space of the form M2 ×R1, where M2 is a con-nected Riemannian surface and M2 × R1 is endowed with the Lorentzian metric〈, 〉 = 〈, 〉M − dt2. In particular, when M is a (necessarily complete) Riemannian sur-face with non-negative Gaussian curvature KM, we prove that any complete maximalsurface in M2 ×R1 must be totally geodesic. Besides, if M is non-flat we concludethat it must be a slice M× t0, t0 ∈ R (here by complete it is meant, as usual, thatthe induced Riemannian metric on the maximal surface from the ambient Lorentzianmetric is complete). We prove that the same happens if the maximal surface is com-plete with respect to the metric induced from the Riemannian product M2 ×R. Thisallows us to give also a non-parametric version of the Calabi-Bernstein theorem for
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entire maximal graphs in M2 ×R1, under the same assumptions on KM. Moreover,we also construct counterexamples which show that our Calabi-Bernstein results areno longer true without the hypothesis KM ≥ 0.
On the other hand, we introduce a local approach to our Calabi-Bernstein results,which is based on some parabolicity criteria for maximal surfaces with non-emptysmooth boundary in M2 × R1. In particular, we derive that every maximal graphover a starlike domain Ω ⊆ M is parabolic. This allows us to give an alternativeproof of the non-parametric version of the Calabi-Bernstein result for entire maximalgraphs in M2×R1. Finally, we also introduce a second local approach to our results,which is given by means of a local integral inequality for the squared norm of thesecond fundamental form of the surface.
The results in this lecture are part of our recent research work developed jointly withAlma L. Albujer, and it can be found in the following references:
References
[1] A. L. Albujer, New examples of entire maximal graphs in H2 × R1, DifferentialGeometry and its Applications 26 (2008), 456–462.
[2] A. L. Albujer and L. J. Alías, A local estimate for maximal surfaces in Lorentzianproduct spaces, Matemática Contemporanea 34 (2008), 1–10.
[3] A. L. Albujer and L. J. Alías, Calabi-Bernstein results for maximal surfaces inLorentzian product spaces, Journal of Geometry and Physics 59 (2009), 620–631.
[4] A. L. Albujer and L. J. Alías, Parabolicity of maximal surfaces in Lorentzian productspaces, Mathematische Zeitschrift 267 (2011), 453–464.
12
Stationary-to-Randers correspondence and convexity
Erasmo Caponio ([email protected])Politecnico di Bari
Stationary-to-Randers correspondence (SRC) is a bijective map between Finsler met-rics of Randers type on a smooth manifold S and conformal standard stationaryLorentzian metrics on S×R. In this talk we focus on some convexity properties ina Randers space (as infinitesimal convexity of a hypersurface, convexity of a largesphere in an asymptotically flat manifold) and on their counterpart by SRC. The talkis based on results contained in the following papers:
References
[1] E. Caponio, M. A. Javaloyes, A. Masiello, “On the energy functional on Finslermanifolds and applications to stationary spacetimes”, Mathematische Annalen, inpress;
[2] E. Caponio, M.A. Javaloyes, M. Sanchez, “On the interplay between LorentzianCausality and Finsler metrics of Randers type”, Revista Matematica Iberoameri-cana, 27/3 (2011) 919–952. arXiv:0903.3501;
[3] R. Bartolo, E. Caponio, A.V. Germinario, M. Sanchez “Convex domains ofFinsler and Riemannian manifolds”, Calculus of Variations and Partial Differen-tial Equations, 40/3-4 (2011) 335 - 356. arXiv:0911.0360;
[4] E. Caponio, M.A. Javaloyes, A. Masiello “Finsler geodesics in the presence ofa convex function and their aplications”, Journal of Physics A, Mathematical andTheoretical, 43/13 (2010), 135207 - 135222. arXiv:0910.3176
and on a work in collaboration with A.V. Germinario and M. Sanchez.
13
Polar actions on symmetric spaces of noncompact type
José Carlos Díaz-Ramos ([email protected])Universidad de Santiago de Compostela
An isometric action on a Riemannian manifold is said to be polar if there exists asubmanifold that meets all the orbits of the action orthogonally; such a submanifoldis called a section. A section is known to be totally geodesic, and if it is flat, the actionis said to be hyperpolar [4].
Polar actions on Euclidean spaces have been classified by Dadok [5]. The classi-fication of hyperpolar actions on symmetric spaces of compact type follows fromthe work by Podestà and Thorbergsson in rank one [10], and by Kollross in higherrank [7]. The classification problem for polar actions on symmetric spaces of com-pact type is still an open problem (see [8]), but it is interesting to emphasize that noexamples of polar, non-hyperpolar actions on symmetric spaces of compact type andrank higher that 2 are known.
Polar and hyperpolar actions on symmetric spaces of noncompact type turn out to bemuch more involved. In some case one can use duality between symmetric spaces ofcompact and noncompact type to derive classification results [6], [9], but this strategydoes not work in general. In fact, there are examples of polar actions in symmetricspaces of noncompact type that have no counterpart in compact type [3]. Some ofthese examples are polar but not hyperpolar.
Some partial classifications of polar and hyperpolar actions can be obtained for sym-metric spaces of noncompact type. The aim of this talk is to present the latest devel-opments in this area [1], [2], [3].
References
[1] J. Berndt, J. C. Díaz-Ramos: Homogeneous polar foliations on complex hyper-bolic spaces, preprint.
[2] J. Berndt, J. C. Díaz-Ramos: Polar actions on the complex hyperbolic plane,preprint.
[3] J. Berndt, J. C. Díaz-Ramos, H. Tamaru: Hyperpolar homogeneous foliations onsymmetric spaces of noncompact type, J. Differential Geom. 86 (2010) 191-235.
[4] L. Conlon: Variational completeness and K-transversal domains, J. DifferentialGeom. 5 (1971), 135–147.
[5] J. Dadok: Polar coordinates induced by actions of compact Lie groups, Trans.Amer. Math. Soc. 288 (1985), 125–137.
[6] J. C. Díaz-Ramos, A. Kollross: Polar actions with a fixed point, Differential Geom.Appl. 29 (2011), 20–25.
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[7] A. Kollross: A classification of hyperpolar and cohomogeneity one actions,Trans. Amer. Math. Soc. 354 (2002), 571–612.
[8] A. Kollross: Polar actions on symmetric spaces, J. Differential Geom. 77 (2007),425–482.
[9] A. Kollross: Duality of symmetric spaces and polar actions,arXiv:1101.1675[math.DG].
[10] F. Podestà, G. Thorbergsson: Polar actions on rank-one symmetric spaces, J.Differential Geom. 53 (1999), 131–175.
The c-boundary of spacetimes and its related boundariesin Differential Geometry.
Jónatan Herrera ([email protected])Universidad de Málaga
In Lorentzian Geometry, the problem of attaching an ideal boundary to any space-time has been one of the most controversial topics along the last decades. Recently,the notion of causal boundary has been consistently redefined and supported by anexhaustive analysis. Moreover, the computation of this causal boundary on any stan-dard stationary spacetime suggests a new completion in Riemannian and Finsleriangeometries. This completion, called Busemann completion has its own interest and canbeen compared with more classical completions, such as the Cauchy one for a metricspace and the Gromov one for a length space.
The aim of this lecture is threefold: (1) to introduce the new notions of c-boundaryand c-completion for strongly causal spacetimes, (2) to introduce the Busemanncompletion and to compare it with the (extension to arbitrary Finsler manifolds of)Cauchy and Gromov’s one, and (3) to compute systematically the c-boundary of any(standard) static and stationary spacetime by using the Busemann boundary.
15
Indefinite extrinsic symmetric spaces
Ines Kath ([email protected])Universität Greifswald
We will study symmetric submanifolds of pseudo-Euclidean spaces. A non-degeneratesubmanifold of a pseudo-Euclidean space is called symmetric submanifold or ex-trinsic symmetric space if it is invariant under the reflection at each of its affinenormal spaces. In particular, each extrinsic symmetric space is an ordinary (abstract)symmetric space. Another characterisation can be obtained in terms of the secondfundamental form. Extrinsic symmetric spaces are exactly those connected completesubmanifolds whose second fundamental form is parallel. While a nice construc-tion found by Ferus provides a classification of all extrinsic symmetric spaces inEuclidean ambient spaces the pseudo-Riemannian situation is much more involved.We will give a description of extrinsic symmetric spaces in pseudo-Euclidean spacesby corresponding innitesimal objects and discuss the classification problem for theseobjects.
Stability of marginally outer trapped surfaces andapplications.
Marc Mars ([email protected])Universidad de Salamanca
Marginally outer trapped surfaces (MOTS) are associated with strong gravitationalfields and generalize the notion of minimal surfaces to a spacetime setting. In thistalk I will describe the notion of stability of marginally outer trapped surfaces and Iwill present several consequences of it. In particular, I will discuss the implications ofstability on the spacetime evolution of MOTS, the existence of outermost MOTS in agiven spacelike hypersurface, the relationship between stable MOTS and symmetriesand an application to a recent area-angular momentum inequality for black holes.
16
How to reconstruct a metric by its unparameterized geodesics
Vladimir S. Matveev ([email protected])Friedrich-Schiller-Universität Jena
We discuss whether it is possible to reconstruct a metric by its unparameterizedgeodesics, and how to do it effectively. We explain why this problem is interestingfor general relativity. We show how to understand whether all curves from a suffi-ciently big family are unparameterized geodesics of a certain affine connection, andhow to reconstruct algorithmically a generic 4-dimensional metric by its unparame-terized geodesics. The algorithm works most effectively if the metric is Ricci-flat. Wealso prove that almost every metric does not allow nontrivial geodesic equivalence,and construct all pairs of 4-dimensional geodesically equivalent metrics of Lorenzsignature. If the time allows, I will also explain how this theory helped to solve twoproblems explicitly formulated by Sophus Lie in 1882, and the semi-Riemanniantwo-dimensional version of the projective Lichnerowicz-Obata conjecture.
The new results of the talk are based on the papers
arXiv:1010.4699, arXiv:1002.3934, arXiv:0806.3169, arXiv:0802.2344, arXiv:0705.3592
joint with Bryant, Bolsinov, Kiosak, Manno, Pucacco
17
New examples of maximal surfaces in Lorentz Minkowski3-Space.
Masaaki [email protected] Institute of Technology
Kotaro [email protected] Institute of Technology
Maximal surfaces in Lorentz Minkowski 3-space arise as solutions of the variationalproblem of locally maximizing the area among spacelike surfaces. By definition,they have everywhere vanishing mean curvature. Like the case of minimal surfacesin Euclidean 3-space, maximal surfaces possess a Weierstrass-type representationformula.
The most significant difference between minimal and maximal surfaces is the factthat the only complete spacelike maximal surfaces are planes. However, if we al-low some sorts of singular points for maximal surfaces, the situation changes. Theauthors showed that if admissible singular points are included, then there is an in-teresting class of objects called "maxfaces", and introduced notions of completenessand weak completeness.
In this talk, we shall survey the theory of maxfaces, and shall introduce several newexamples.
18
Minicourses
Causality and Legendrian linking
Vladimir Chernov ([email protected])Dartmouth College
Given a globally hyperbolic spacetime (Xm+1, g) put N to be the space of futuredirected light rays (null geodesics) in X. The sphere of all light rays through x ∈ Xis called the sky Sx of x. Low observed that N is a contact manifold and Sx isa Legendrian sphere. Low and later Natario and Tod conjectured that for manyspacetimes causal relations between x, y ∈ X can be formulated in terms of linkingof the pair of skies (Sx, Sy).
Nemirovski and myself showed that when the universal cover of the Cauchy surfaceof X is an open manifold, one has that two events x, y ∈ X are causally related ifand only if the Legendrian link (Sx, Sy) is nontrivial. Low, and later Natario andTod, conjectured that for many spacetimes, causal relations between x, y ∈ X can beformulated in terms of linking of the pair of skies (Sx, Sy).
Poincaré conjecture, proved by Perelman and combined with the above results, im-plies that for 3 + 1-dimensional spacetimes, linking completely determines causalityunless the universal cover of the Cauchy surface is S3. Linking does not determinecausality in refocussing spacetimes and we discuss relation between refocussing andthe Yx
l -manifolds studied by Bérard-Bergery.
19
On the isometry group of Lorentz manifolds
Paolo Piccione ([email protected])Universidade de São Paulo
The course will consist of three lectures. In the first lecture I intend to discuss a proofof the existence of a Lie group structure in the set of isometries, or more generally ofconformal diffeomorphisms, of a semi-Riemannian manifold. In the second lectureI will present the main results of a theory developed mostly by Adam, Stuck andZeghib in the 90’s, that lead to a classification of Lie groups that act locally faithfullyand isometrically on compact Lorentzian manifolds. Finally, in the third lecture, Iwill present some recent result that have been obtained by Zeghib and myself onthe isometry group of compact Lorentz manifolds that admit a somewhere timelikeKilling vector field.
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Short talks
Uniqueness and non-existence results for spacelikehypersurfaces with constant mean curvature in Hn ×R1
Alma L. Albujer ([email protected])Universidad de Córdoba
In this talk we present some uniqueness and non-existence results for completespacelike hypersurfaces with constant mean curvature in the Lorentzian productspace Hn ×R1. In order to obtain our results we ask the normal hyperbolic angle ofthe hypersurface to be bounded in an appropriate way. We get our results as a conse-quence of the well known Omori-Yau generalized maximum principle for completeRiemannian manifolds.
On the other hand it is well known that a spacelike entire graph in Hn ×R1 is notnecessarily complete. However, we are also able to state our results in terms of entiregraphs. That is, we present non-parametric versions of our results.
This talk is part of joint work with Fernanda E. C. Camargo and Henrique F. deLima.
References
[1] A. L. Albujer, New examples of entire maximal graphs in H2 ×R1, DifferentialGeom. Appl. 26 (2008), 456–462.
[2] A. L. Albujer, F. E. C. Camargo and H. F. de Lima, Complete spacelike hyper-surfaces with constant mean curvature in −R ×Hn, J. Math. Anal. Appl. 368(2010), 650–657.
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On the space of geodesics of Riemannian and Lorentzianspace forms
Henri Anciaux ([email protected])Universidade de São Paulo
The space of geodesics of a given type (timelike or spacelike) L±(Sn+1p ) of a non-flat
pseudo-Riemannian space form Sn+1p of signature (p, n + 1 − p) and dimension n
enjoys a natural invariant structure: in the case of periodic geodesics, it is (pseudo)-Kähler, while in the case of unbounded geodesics, it enjoys a para-Kähler struc-ture.
We prove that the underlying metric of L±(Sn+1p ) has constant scalar curvature, and
is moreover Einstein if and only if Sn+1p is Riemannian (thus the sphere or the hyper-
bolic space) or Lorentzian (thus the Sitter or anti de Sitter spaces), and the geodesicsare timelike.
We also study the normal congruence of a hypersurface S of Sn+1p , which happens
to be a Lagrangian submanifold S of L±(Sn+1p ), and relate the geometries of S and
S. In particular S is totally geodesic if and only if S has parallel second fundamentalform. In the Riemannian and spacelike Lorentzian cases, we express the minimalityof S by a functional relation involving the principal curvatures of S..
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Geodesic connectedness on Gödel type spacetimes: a “static”variational set-up
Rossella Bartolo ([email protected])Politecnico di Bari
We show that the problem of the geodesic connectedness of a wide class of Gödeltype spacetimes can be reduced to the search of critical points of a functional nat-urally involved in the study of geodesics in standard static spacetimes. Then, theaccurate estimates in [2] allow us to improve substantially the result in [3], by aweakening of the boundedness assumptions about the metric coefficients.
References
[1] R. Bartolo, A.M. Candela, J.L. Flores, A note on geodesic connectedness in Gödeltype spacetimes, preprint, 2011.
[2] R. Bartolo, A.M. Candela, J.L. Flores, M. Sánchez, Geodesics in static Lorentzianmanifolds with critical quadratic behavior, Adv. Nonlinear Stud. 3 (2003), 471–494.
[3] A.M. Candela, M. Sánchez, Geodesic connectedness in Gödel type space-times,Differential Geom. Appl. 12 (2000), 105–120.
Characterization and structure of second-order symmetricLorentzian manifolds
Oihane F. [email protected] de Granada
José M.M. [email protected] del PaísVasco
Miguel Sánchez [email protected] de Granada
Recently (arXiv:1101.5503v2), we have characterized the second-order symmetricLorentzian manifolds (i.e., those which satisfy ∇2Rie = 0). In particular, all thecomplete simply-connected non-locally symmetric ones, constitute an extension ofthe Cahen-Wallach family of (n-dimensional) plane waves. Here, we will sketch thetechniques for both, local and global results.
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On one class of holonomy groups in pseudo-Riemanniangeometry
Alexey [email protected] University
Dragomir [email protected] University
Holonomy groups were introduced by Élie Cartan [3] in 1926 in order to study theRiemannian symmetric spaces and since then the classification of holonomy groupshas remained one of the classical problems in differential geometry.
Definition. Let M be a smooth manifold and ∇ an affine connection on TM. The holonomygroup of ∇ is a subgroup Hol(∇) ⊂ GL(Tx M) that consists of the linear operators A :Tx M → Tx M being “parallel transport transformations” along closed loops γ with γ(0) =γ(1) = x.
In the case of Levi-Civita connections on Riemannian manifolds, the classificationof holonomy groups is due to M. Berger [1]. In the pseudo-Riemannian case, thedescription of holonomy groups is a very difficult problem which still remains openand even particular examples are of interest. We refer to [4], [5] for more informationon recent development in this field.
Since we deal with Levi-Civita connections only, we consider subgroups of the(pseudo)-orthogonal group SO(g), where g is a non-degenerate inner product ona finite-dimensional real vector space V. The main result of our paper is
Theorem. For every g-symmetric operator L : V → V, the identity component of itscentraliser in SO(g)
GL = X ∈ SO(g) | XL = LXis a holonomy group for some (pseudo)-Riemannian metric.
In the Riemannian case this theorem becomes trivial as L is diagonalisable and GLis isomorphic to the direct product SO(k1) × · · · × SO(km) ⊂ SO(n), which is, ofcourse, a holonomy group. In the pseudo-Riemannian case, L may have non-trivialJordan blocks and the structure of GL becomes more complicated.
As usual, instead of GL we consider its Lie algebra gL. The first step is to prove thatgL is a Berger algebra. To that end, it is sufficient to construct a formal curvaturetensor R : Λ2(V) → so(g) such that the image of R coincides with gL. The next stepis to realise gL as a holonomy Lie algebra for a suitable metric.
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Our proof is based on unexpected relationship between curvature tensors for somespecial metrics and the theory of integrable systems on semi-simple Lie algebras (see[2]) . The following two “magic formulas” play crucial role in our construction:
R(X) =ddt∣∣t=0 pmin(L + t · X), X ∈ so(g) ' Λ2V, (*)
C = R(⊗) = ddt∣∣t=0 pmin(L + t · ⊗), (**)
where pmin(λ) is the minimal polynomial of L. The first formula gives a map R :Λ2V → gl(V). The second one defines a tensor C of type (2, 2) whose meaning canbe understood from the following example: if pmin(λ) = λm, then
R(X) =m
∑k=1
Lk−1XLm−k and C = R(⊗) =m
∑k=1
Lk−1 ⊗ Lm−k
These two algebraic objects possess the following remarkable properties:
Proposition. R satisfies the Bianchi identity and its image is contained in gL. In otherwords, R is a formal curvature tensor for the Lie algebra gL.
So far L and g were defined on a fixed vector space V which now will be consideredas T0Rn. We now extend them onto a neighborhood of 0 ∈ Rn.
Proposition. In local coordinates (x1, . . . , xn), we set L = L(0) = const and
g = gij(x) = gij(0) +12 ∑ giα(0)gpβ(0)C
α,βj ,q xpxq
Then L is g-symmetric,∇L = 0 and the curvature tensor for g at the origin 0 ∈ Rn is R(X)defined by (*).
Using a kind of “block-wise” modification of formulas (*) and (**), it is not hard toconstruct a formal curvature tensor R(X) and (2, 2)-tensor C for which Propositions1 and 2 still hold, but R is, in addition, an “onto” map. This obviously completes theproof.
References
[1] M. Berger, Bull. Soc. Math. France, 83 (1955), 279–330.[2] A. Bolsinov, V. Kiosak, V. Matveev, Jour. London Math. Soc., 80(2009), 341–356.[3] É. Cartan, Acta.Math. 48 (1926), 1-42 .[4] A. Galaev, arXiv:0906.5250, to appear in: Lobachevskii J. Math. 32 (2011), no. 2.[5] T. Leistner, J. Diff. Geom. 76 (2007), no. 3, 423–484.
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Uniqueness of spacelike hypersurfaces of constant meancurvature in cosmological models with certain symmetries
Magdalena Caballero ([email protected])Universidad de Córdoba
In general relativity, symmetry is usually based on the assumption of the existence ofa one-parameter group of transformations generated by a Killing or, more generally,conformal vector field. Although it is not always assumed the same causal characterfor the infinitesimal symmetry, the timelike choice is natural, since the integral curvesof such a timelike infinitesimal symmetry provide a privileged class of observers inthe spacetime. The presence of such a vector field is not enough to prevent the ex-istence of closed causal curves. However, if the timelike conformal vector field isglobally the gradient of some smooth function, then the (clearly noncompact) space-time admits a global time function. Therefore, it is stably causal. Finally, spacetimeswith a timelike gradient conformal vector field (GCS spacetimes) have another inter-esting property, they admit a foliation by constant mean curvature (CMC) spacelikehypersurfaces.
The aim of this talk is to introduce a new technique to study CMC spacelike hyper-surfaces in GCS spacetimes and to give new uniqueness results for compact CMCspacelike hypersurfaces in these ambient spacetimes, both in the parametric andnonparametric case. As an application, the leaves of the natural spacelike foliation ofsuch spacetimes are characterized in some relevant cases.
This talk is based on a joint work with Alfonso Romero and Rafael M. Rubio.
References
[1] M. Caballero, A. Romero and R. M. Rubio, Constant mean curvature space-like hypersurfaces in Lorentzian manifolds with a timelike gradient conformalvector field, Classical and Quantum Gravity (2011), to appear.
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Pinching the Fermat metric in stationary spacetimes
Alexander Dirmeier ([email protected])Technische Universität Berlin
Given a standard stationary spacetime and a fixed foliation by spacelike slices, thereis a canonical Finsler metric of Randers type on the spacelike hypersurfaces, whichis called the Fermat metric. The spacelike slices are Cauchy hypersurfaces iff theFermat metric is (forward and backward) complete. If one can find Riemannian met-rics smaller than the Fermat metric on all the slices, their completeness imply thecompleteness of the Fermat metric and, hence, global hyperbolicity of the spacetime.Accordingly, given a globally hyperbolic stationary spacetime and apt Cauchy hyper-surfaces as spacelike slices, this ensures the completeness of all Riemannian metricson the slices which are bigger than the Fermat metric. Smaller and bigger is under-stood here in the sense of a half-order; e.g. for a Riemannian metric g and a Randersmetric F on manifold M we have
√g ≤ F :⇔
√gx(v, v) ≤ F(x, v), ∀x ∈ M, v ∈ Tx M.
We derive several smaller and bigger Riemannian metrics, which are optimal in thesense of finding the biggest metric, that is smaller or the smallest metric that is big-ger in a specific situation. Therefore, we pinch the Fermat metric by Riemannianmetrics. The new metrics arise from conformal transformations of some canonicalchoices of Riemannian metrics on the slices. This results in several new necessaryand sufficient conditions for the slices to be Cauchy hypersurfaces, based on Rie-mannian completeness and the growth of functions on the hypersurfaces. Especiallywe are able to give a sufficient condition for global hyperbolicity only based on thegrowth of a function. Hence, this constitutes a purely analytic criterion for globalhyperbolicity. For more details see [1].
References
[1] A. Dirmeier, M. Plaue, and M. Scherfner, Growth Conditions, Riemannian Com-pleteness and Lorentzian Causality, to appear in J. Geom. Phys. (2011).
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Families of parallel CMC hypersurfaces in noncompactrank-one symmetric spaces
Miguel Domínguez-Vá[email protected] Santiago de Compostela
José Carlos Dí[email protected] Santiago de Compostela
In any Riemannian ambient manifold, a hypersurface is said to be isoparametric if itand its nearby equidistant hypersurfaces have constant mean curvature. The study ofthese families of parallel CMC hypersurfaces was started by B. Segre and É. Cartan,who classified these objects in Euclidean and real hyperbolic spaces. In constrast towhat happens in these two cases, in spheres there exist isoparametric hypersurfaceswhich are not homogeneous (that is, orbits of an isometric action). This is one ofthe reasons why the problem in spheres is much more involved and has motivatedmany interesting ideas in the last decades.
A remarkable feature of isoparametric hypersurfaces in real space forms is that theyhave constant principal curvatures. Under certain additional hypotheses, this relationstill holds for more general ambient spaces, such as nonflat complex space forms.This allows us to get some partial classifications in complex space forms, based onthe results achieved in [2].
However, in general, an isoparametric hypersurface does not have constant principalcurvatures. Many examples of this behaviour were found by the authors in [1], forthe case of complex hyperbolic spaces.
In this communication, we will explain the extension of the construction in [1] tononcompact rank-one symmetric spaces. As a by-product, we will obtain the firstknown inhomogeneous isoparametric hypersurface with constant principal curva-tures in a space different from a sphere.
References
[1] J. C. Díaz-Ramos, M. Domínguez-Vázquez, Inhomogeneous isoparametric hy-persurfaces in complex hyperbolic spaces, preprint, 2010, arXiv:1011.5160v1. Toappear in Math. Z.
[2] J. C. Díaz-Ramos, M. Domínguez-Vázquez, Non-Hopf real hypersurfaceswith constant principal curvatures in complex space forms, preprint, 2009,arXiv:0911.3624v1. To appear in Indiana Univ. Math. J.
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Totally geodesic submanifolds in Robertson-Walker spaceswith flat fibers
Zdenek Duš[email protected] University
Miguel [email protected] de Granada
We determine all totally geodesic submanifolds of Robertson-Walker spaces −I × fRn. We write down the differential equations for a totally geodesic submanifold ofsuch a space and, in the nontrivial case f ′ 6= 0, we derive the necessary condition forthe function f . We solve explicitly the corresponding systems of equations.
References
[1] Z. Dušek, M. Ortega, Totally geodesic submanifolds in Robertson-Walker spaces withflat fibers, in preparation.
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Lorentzian Bobillier Formula
Soley [email protected] University
Nurten [email protected] Teknik Üniversitesi
In this paper, Lorentzian Euler Savary formula (giving the relation between the cur-vatures of the trajectory curves drawn by the points of the moving plane in the fixedplane) during one parameter Lorentzian planar motion is taken into consideration.By using an original geometrical interpretation of Lorentzian Euler Savary formula,Lorentzian Bobillier formula is established.
However, another presentation is made in this paper without using the Euler Savaryformula. Then the Lorentzian Euler Savary formula will appear as a particular caseof Bobillier formula and as a result of the direct way chosen, this new Lorentzianformula (Bobillier) could be considered as a fundamental law in a planar Lorentzianmotion in place of Euler Savary’s.
References
[1] M. Fayet, Une Nouvelle Formule Relative Aux Courbures Dans un MouvementPlan, Mech. Mach. Theory 23(2), (1988) 135–139.
[2] E.A. Dijskman, Motion geometry of mechanism, Cambridge University Press, Cam-bridge, 1976.
[3] M. Fayet, Bobillier Formula as a Fundamental Law in Planar Motion, Z. Angew.Math. Mech., 82(3), (2002) 207–210.
[4] N. Rosenauer and A.H. Willis, Kinematics of Mechanisms, General PublishingCompany, Ltd., Toronto, Ontario, 1953.
[5] A.A. Ergin, On the one–parameter Lorentzian motion, Comm. Fac. Sci. Univ.,Series A, 40, (1991) 59–66.
[6] M. Ergüt, A. P. Aydın, and N. Bildik, The Geometry of Canonical Relative Sys-tems and One-Parameter Motions in 2-Lorentzian Space, The Journal of FiratUniversity, 3, (1988) 113–122.
[7] T. Ikawa, Euler-Savary’s Formula on Minkowski Geometry, Balkan J. Geom. Appl.,8(2), (2003) 31–36.
[8] M.A. Güngör, A.Z. Pirdal, and M. Tosun, Euler–Savary Formula for theLorentzian Planar Homothetic Motions, Int. J. Math. Comb. 2, (2010) 102–111.
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Lorentzian quasi-Einstein manifolds
S. Gavino-Fernández ([email protected])Universidad Santiago de Compostela
A pseudo-Riemannian manifold (M, g) of dimension n + 2, n ≥ 1, is quasi-Einstein ifthere exists a smooth function f : M→ R such that
ρ + Hes f − µd f ⊗ d f = λg,
where ρ and Hes f are the Ricci tensor and the Hessian of f , for some constantsµ, λ ∈ R [1], [2]. If the function f is constant we get the Einstein equation and ifµ = 0 we obtain the gradient Ricci soliton equation [3]. Moreover, the existence ofquasi-Einstein metrics is closely related to the existence of warped product Einsteinmetrics. Indeed, if M × f F is Einstein, then φ = −(dimF) ln f is a quasi-Einsteinstructure: ρ + Hesφ − 1
dimF dφ⊗ dφ = λg.
The aim of this work is to investigate locally conformally flat quasi-Einstein Lorentzianmanifolds, showing that they are locally isometric to a space form, a warped productof Robertson-Walker type or a locally conformally flat pp-wave.
References
[1] G. Catino, C. Mantegazza, L. Mazzieri and M. Rimoldi, Locally conformally flatquasi-Einstein manifolds, arXiv:1010.1418v3 [math.DG].
[2] Jeffrey S. Case, Singularity theorems and the Lorentzian splitting theorem forthe Bakry-Emery-Ricci tensor, J. Geom. Phys. 60 (2010), no. 3, 477-490.
[3] M. Brozos-Vázquez, E. García-Río and S. Gavino-Fernández, Locally confor-mally flat Lorentzian gradient Ricci solitons, arXiv:1106.2924v1 [math.DG].
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The geometry of collapsing isotropic fluids
Roberto Giambò ([email protected])Università degli Studi di Camerino
The study of spacetimes modeling a collapsing spherical isotropic fluid has alwaysbeen a recurrent topic for relativists – its connection with Tolman–Bondi solution,one of the few known-in-details solutions dynamically collapsing to a singularity,makes it one the most intriguing problem in gravitational collapse. Some results areknown from numerical relativity but very few is known about the geometry of thesolutions – whether a singularity is developed, and if that is the case, what is thecausal structure of the solution. I will sketch a recent line of research aiming to shednew light on the above aspects.
Joint research (work in progress) with: Fabio Giannoni (Camerino), Giulio Magli (Mi-lan Politecnico), Daniele Malafarina and Pankaj S Joshi (Tata Institute, Mumbai)
References
[1] R. Giambò, F. Giannoni, P. Piccione, G. Magli, Commun Math Phys 235(3) (2003),545-563, and Class. Quantum Grav., 20 (2003) L75-L82
[2] P S Joshi and D Malafarina, Phys Rev D83 064007 (2011) (with M Patil) and PhysRev D83 024009 (2011)
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Kähler and para-Kähler Weyl manifolds of dimension 4
P. [email protected] of Oregon
M. [email protected] A Coruña
S. [email protected] InstituteSano
Let (M, g) be a pseudo-Riemannian manifold. Let J± be an integrable almost (para)-complex structure, i.e. an endomorphism of the tangent bundle so that J2
± = ±idand so that the associated Nijenhuis tensor vanishes. One assumes J∗±g = ∓g; in thepara-complex setting necessarily g has neutral signature. The triple (M, g, J±) is thensaid to be a para/pseudo-Hermitian manifold.
A torsion free connection ∇ on the tangent bundle of M is said to be a Weyl con-nection and (M, g,∇) is said to be a Weyl structure if there is a 1-form φ so that∇g = −2φ ⊗ g. We examine the interaction of these two structures. One says that(M, g, J±,∇) is a para/pseudo-Kähler Weyl manifold if ∇J± = 0 (this conditionautomatically implies J± is integrable). In dimension m ≥ 6, the Weyl structureof any para/pseudo-Kähler Weyl manifold is trivial, i.e. there exists a conformallyequivalent metric g so that (M, g, J±) is Kähler and so that ∇ is the Levi-Civita con-nection determined by g. The 4-dimensional setting is very different and forms thefocus here: one has that every pseudo-Hermitian manifold of dimension 4 admits aunique Kähler Weyl structure and, similarly, that every para-Hermitian manifold ofdimension 4 admits a unique para-Kähler Weyl structure. These results rely upona curvature decomposition which extends previous results of Higa to the complexsetting and also upon an extension of the classical Gray-Hervella decomposition tothe indefinite setting.
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Extrinsically flat surfaces of space forms and the geometricstructure on the space of oriented geodesics
Atsufumi Honda ([email protected])Tokyo Institute of Technology
It is well known that any complete extrinsically flat surface in the 3-sphere S3 must betotally geodesic, where “extrinsically flat” means vanishing of the Gauss-Kroneckercurvature. However, if one admits some singularities, there are many non-trivialcomplete extrinsically flat surfaces in S3. In this talk, we shall introduce a representa-tion formula for extrinsically flat surfaces in S3 in terms of a pair of two curves in the2-sphere S2.
The main idea for the representation formula is the correspondence between ruledsurfaces in S3 and curves in L(S3), where L(S3) is the space of oriented geodesicsin S3. Then we use the metric on L(S3) associated to the minitwistor complex struc-ture.
We shall also introduce some related results for anti-de Sitter 3-space H31 .
References
[1] N. J. Hitchin, Monopoles and geodesics, Comm. Math. Phys. 83 (1982), 579–602.[2] A. Honda, Isometric Immersions of the Hyperbolic Plane into the Hyperbolic
Space, preprint, 2010, arXiv:1009.3994.
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Spacelike hypersurfaces of constant higher order meancurvature in generalized Robertson-Walker spacetimes
Debora Impera ([email protected])Università degli Studi di Milano
Spacelike hypersurfaces in spacetimes are objects of increasing interest in recentyears, both from a physical and a mathematical point of view. A basic questionon this topic is the problem of uniqueness of spacelike hypersurfaces with constantmean curvature in certain spacetimes, and, more generally, that of spacelike hyper-surfaces with constant higher order mean curvature.
In a recent paper, Alías and Colares [1] studied in depth the problem of unique-ness for compact spacelike hypersurfaces with constant higher order mean curva-ture in spatially closed generalized Robertson-Walker spacetimes. Their approachwas based on the use of the so called Newton transformations Pk and their asso-ciated second order differential operators Lk, as well as on the application of somegeneral Minkowski integral formulae for compact hypersurfaces. In this talk, whichis based on the work in [3], we go deeper into this study. We consider first the caseof compact spacelike hypersurfaces, completing some previous results given in [1]and we next extend these results to the complete noncompact case. Our approach isbased on the use of a generalized version of the Omori-Yau maximum principle fortrace type differential operators which includes the operators Lk that has been re-cently introduced by the authors in [2] for the study of hypersurfaces in Riemannianwarped products.
This is a joint work with L. J. Alías and M. Rigoli.
References
[1] L. J. Alías and A. G. Colares, Uniqueness of spacelike hypersurfaces with con-stant higher order mean curvature in generalized Robertson-Walker spacetimes,Math. Proc. Cambridge Philos. Soc. 143 (2007), 703–729
[2] L. J. Alías, D. Impera and M. Rigoli, Hypersurfaces of constant r + 1-mean cur-vature in warped product spaces, preprint.
[3] L. J. Alías, D. Impera and M. Rigoli, Spacelike hypersurfaces of constant higherorder mean curvature in generalized Robertson-Walker spacetimes, preprint.
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Area-Angular momentum inequality in stable marginallytrapped surfaces
José Luis [email protected] fürGravitationsphysik
Martín [email protected] fürGravitationsphysik
Sergio [email protected] Nacional deCórdoba, Argentina
We discuss the area-angular momentum inequality A ≥ 8π|J| and show that it holdsfor axially symmetric closed outermost stably marginally trapped surfaces. Such sur-faces are sections of dynamical trapping horizons, namely hypersurfaces that pro-vide quasi-local models of black hole horizons in otherwise generic Lorentzian man-ifolds whose Einstein tensor satisfies a dominant energy condition. This inequalityrepresents a particular example of the extension to a Lorentzian setting of tools em-ployed in the discussion of minimal surfaces in a Riemaniann context.
K-conformal maps of a stationary spacetime and Randersmetric
Miguel A. [email protected] de Murcia
Leandro [email protected] de São Paulo
Miguel Sá[email protected] de Granada
Paolo [email protected] de São Paulo
We study the conformal maps of a stationary spacetime fixing the flow lines of thetimelike Killing field K, which we call K-conformal maps. We will use the relationbetween a stationary spacetime (R× S, g) and a Randers metric in the base S to studythese maps. In particular, K-conformal maps project to maps in S that preserves theRanders metric up to the differential of a function. We will deduce some resultsabout the genericity of these maps.
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Classification of symmetric M-theory backgrounds
Figueroa O’Farrill ([email protected])University of Edinburgh
We classify symmetric backgrounds of eleven-dimensional supergravity up to localisometry. In other words, we classify triples (M, g, F), where (M, g) is an eleven-dimensional lorentzian locally symmetric space and F is an invariant 4-form, satisy-ing the equations of motion of eleven-dimensional supergravity. The possible (M, g)are given either by (not necessarily nondegenerate) Cahen–Wallach spaces or byproducts AdSd ×M11−d for d = 2, ..., 7 and M11−d a not necessarily irreducible rie-mannian symmetric space. For each such geometry we determine the possible Fs.The backgrounds are grouped in families sharing the same F-moduli space, whichwe determine completely in most cases.
On the geometric structure of Ori spacetimes
Mike Scherfner ([email protected] )Technische Universität Berlin
Starting in 1993 A. Ori published three spacetimes violating the chronology con-dition, being close to physical reality in some sense. We would like to reveal thedifferential geometric structure of these models, i. e. CTCs in detail, geodesics andsymmetry and we will construct a new model of Ori-type starting from a Kerr metric.The origin of this talk is an article together with J. Dietz and A. Dirmeier.
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Pseudo-umbilical and related surfaces in spacetime
José M. M. Senovilla ([email protected])Universidad del País Vasco
Consider a spacelike surface S imbedded in a 4-dimensional Lorentzian manifold(V4, g). S will be said to be umbilical along a direction ~N normal to S if the secondfundamental form along that direction is proportional to the first fundamental formof S. In particular, S is pseudo-umbilical if it is umbilical along the mean curva-ture vector field, and (totally) umbilical if it is umbilical along all possible normaldirections.
I will prove that the necessary and sufficient condition for S to be umbilical along anormal direction is that two independent Weingarten operators (and, a fortiori, allof them) commute, or equivalently that the shape tensor be diagonalizable on S. Theumbilic direction is then uniquely determined.
This can be seen to be equivalent to a condition relating the normal curvature andthe appropriate part of Riemann tensor of the spacetime. In particular, for confor-mally flat spacetimes (including Lorentz space forms) the neccessary and sufficientcondition is that the normal curvature vanishes.
Some furhter consequences will also be analyzed.
Splitting Theorems in Lorentzian Geometry
Didier A. Solis ([email protected])Universidad Autónoma de Yucatan
According to a general principle in comparison theory, the existence of a line -that is,a causal globally maximizing and complete geodesic- is incompatible with standardenergy conditions, except in very special cases. Thus, the geometry of a spacetime inwhich a line can be formed without violating appropriate energy conditions is veryspecial. This is the basis of the so called Splitting Theorems in Lorentzian Geometry.Of special interest are those splitting theorems in which the hypothesis are enoughto guarantee constant curvature of spacetime. In this talk we present an overview ofthe classical results as well as recent developments in the context of asymptoticallyflat and asymptotically de Sitter spacetimes.
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Closed Geodesics in Lorentzian Surfaces
Stefan Suhr ([email protected])Universität Hamburg
G. Galloway proved in [1] that every closed Lorentzian surface contains at least oneclosed timelike or null geodesic. From the perspective of dynamical systems this re-sult is not optimal. This is because the null geodesics need not be be closed geodesicsin the usual sense, i.e. the tangent curves are not closed orbits of the geodesic flow.This is due to the fact that γ(T) ∼ γ(0) does not imply γ(T) = ±γ(0) if γ is alightlike geodesic. This leaves the possibility of closed Lorentzian surfaces withoutclosed geodesics, at least from a dynamical point of view.
My communication will give an overview of the proofs of the following results:
Theorem ([2]). Every closed (i.e. compact and with empty boundary) Lorentzian surfacecontains two closed geodesics, one of which is definite, i.e. time- or spacelike. This lowerbound is optimal.
Theorem ([2]). The geodesic flow of a closed Lorentzian surface has at least two closedorbits. This lower bound is optimal.
Cororally ([2]). Every time and space orientable closed Lorentzian surface contains fourclosed geodesics, two of which must be definite, i.e. every spacetime structure on the 2-toruscontains at least four closed geodesics.
Examples attaining the lower bounds will be given. I will conclude by discussingthe relationship between the least number of closed geodesics in a closed Lorentziansurface in comparison to the connected component in the space of all Lorentzianmetrics on that surface.
References
[1] G. Galloway, Compact Lorentzian Manifolds without Closed NonspacelikeGeodesics, Proc. Amer. Math. Soc., 98, (1986), 119–123.
[2] S. Suhr, Closed Geodesics in Lorentzian Surfaces, preprint, 2010,arXiv:1011.4878.
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Conformally flat homogeneous Lorentzian manifolds
Kazumi Tsukada ([email protected])Ochanomizu University
This is a joint work with Kyoko Honda (Ochanomizu University).
We consider the problem to classify conformally flat homogeneous semi-Riemannianmanifolds.Conformally flat homogeneous Riemannian manifolds were classified byH. Takagi [1].They are all symmetric spaces. While three-dimensional conformallyflat homogeneous Lorentzian manifolds were classified by us [2] and the exampleswhich are not symmetric spaces were found. In this talk, we report the result onthe classification of higher dimensional conformally flat homogeneous Lorentzianmanifolds.
Let Mn be an n(≥ 4) dimensional conformally flat homogeneous Lorentzian man-ifold. Our classification depends on the form of the modified Ricci operators A =
1n−2
(Q− S
2(n−1) Id)
, where Q is the Ricci operator and S is the scalar curvature ofM, respectively.
Theorem. Let Mn be an n(≥ 4) dimensional conformally flat homogeneous Lorentzianmanifold. Then the linear operator A has exactly one of the following forms:
Case 1 λIr − λIn−rCase 2 λ(e∗1 ⊗ e2 − e∗2 ⊗ e1 + ∑m+2
i=3 e∗i ⊗ ei −∑2m+2i=m+3 e∗i ⊗ ei). (λ > 0)
Case 3 λIn + εe∗2 ⊗ e1 (λ ≤ 0, ε = ±1 )Case 4 λIr − λIn−r + e∗1 ⊗ e3 + e∗3 ⊗ e2 (3 ≤ r ≤ n, λ < 0)
where Ir = ∑ri=1 e∗i ⊗ ei and In−r = ∑n
i=r+1 e∗i ⊗ ei. Our expressions are those with respectto an orthonormal basis
⟨e1, e1
⟩= −1,
⟨ei, ej
⟩= δij (i, j ≥ 2) in Case 1 and those with
respect to a semi-orthonormal basis⟨e1, e2
⟩= 1,
⟨ei, ej
⟩= δij (i, j ≥ 3) in Cases 2,3, and
4. We denote by e∗1 , e∗2 , · · · , e∗n the dual basis of e1, e2, · · · , en.
We construct examples and classify conformally flat homogeneous Lorentzian man-ifolds for each case.
References
[1] H.Takagi,Conformally flat Riemannian manifolds admitting a transitive group ofisometries,Tohoku Math. J., 27(1975), 103-110.
[2] K.Honda and K.Tsukada,Three-dimensional conformally flat homogeneousLorentzian manifolds,J.of Phys A:Math.Theor.,40(2007),831-851.
40
Slant geometry of spacelike hypersurfaces in the lightconewith respect to the φ-hyperbolic duals
Handan Yıldırı[email protected] University
Shyuichi [email protected] University
One-parameter families of Legendrian dualities which are the extensions of fourdualities obtained in [5] for pseudo-spheres in Lorentz-Minkowski Space were givenin [6]. In this talk, as an application of such extensions, a new extrinsic differentialgeometry on spacelike hypersurfaces in the lightcone is constructed with respect tothe φ-Hyperbolic Duals [7].
References
[1] V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differ-entiable Maps, Vol. I, Birkhäuser, 1986.
[2] M. Asayama, S. Izumiya, A. Tamaoki and H. Yıldırım, Slant geometryof spacelike hypersurfaces in Hyperbolic space and de Sitter space, RevistaMatemática Iberoamericana, accepted, (2011).
[3] A. C. Asperti and M. Dajczer, Conformally flat Riemannian manifolds as hy-persurfaces of the lightcone, Canad. Math. Bull. 32 (1989), 281–285.
[4] L. Chen and S. Izumiya, A mandala of Legendrian dualities for pseudo-spheresin semi-Euclidean space, Proceedings of the Japan Academy 85, Ser. A, (2009), 49–54.
[5] S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone,Moscow Mathematical Journal 9 (2009), 325–357.
[6] S. Izumiya and H. Yıldırım, Extensions of the mandala of Legendrian dualitiesfor pseudo-spheres in Lorentz-Minkowski space, Topology and Its Applications,accepted, (2011).
[7] S. Izumiya and H. Yıldırım, Slant geometry of spacelike hypersurfaces in thelightcone, Journal of the Mathematical Society of Japan, in press, (2011).
41
Posters
Beltrami-Meusnier Formulas of Semi Ruled Surface
Mahmut [email protected] University
Soley [email protected] University
Murat [email protected] University
In this paper, we study the sectional curvatures of the generalized semi ruled sur-faces in semi Euclidean space En+1
v . The first fundamental form and the metriccoefficients of the generalized semi ruled surfaces are calculatedand and in theseregards, Riemannian-Christoffel curvatures are obtained by the help of ChristoffelSymbols. So, the curvatures of arbitrary nondejenere tangential sections of the gen-eralized semi ruled surface is investigated. In addition to this, the relations betweenthe sectional curvatures of semi ruled surfaces are obtained. These are called semi-Euclidean Beltrami-Meusnier formulas.
References
[1] Ekici, C., Görgülü, A., On the curvatures (k + 1)−dimensional semi-ruled sur-faces in En+1
v , Math. Comput. Appl., 3 (2000), 139–148.[2] Ekici, C., Görgülü, A., On The Ricci curvature tensor of (k + 1)−dimensional
semi-ruled surfaces in En+1v , Math. Comput. Appl., 3 (2000), 149–155.
[3] Ersoy, S., Tosun, M., Sectional Cuvature of Timelike Ruled Surface Part I:Lorentzian Beltrami-Euler Formula, Iran. J. Sci. Technol. Trans. A Sci. 34 (2010),197–214.
[4] Ersoy, S., Tosun, M., Lorentzian Beltrami-Euler formula and generalizedLorentzian Lamarle formula in Rn
1 , Le Matematiche, 1 (2009), 25–45.[5] Frank, H., Giering, O., Zur schnittkrümmung verallgemeinerter regelflachen,
Arch. Math. , 1 (1979), 86–90.[6] Frank, H., Giering, O., Verallgemeinerte regelflachen, Math. Z., 150 (1976), 261-
271.[7] Frank, H., Giering, O., Verallgemeinerte Regelflachen im Grassen II, Journal of
Geometry, 23 (1984), 128-140.[8] O’Neill, B., Semi-Riemannian geometry, Academic Press, New York, 1983.[9] Thas, C., Properties of ruled surfaces in the Euclidean space En, Bull. Inst. Math.
Acad. Sinica, 6 (1978), 133–142.
42
[10] Tosun, M., KURUOÐLU, N., On (k+1)-dimensional time-like ruled surface inthe Minkowski space , Institute of Mathematics and Computer Sciences, 11 (1998),1-9.
[11] Ratcliffe, J.G., Foundations of Hyperbolic Manifolds, Department Mathematics,Vanderbilt University, 1994.
About a new type of null Osserman condition on LorentzS-manifolds
Letizia Brunetti ([email protected])Università degli studi di Bari
The Osserman conjecture, introduced in [3] for Riemannian manifolds, relates theproperties of the Riemannian curvature to the spectral behaviour of the Jacobi oper-ator. The Ossermann problem has been partially solved in the Riemannian case and,while it still remains open in the semi-Riemannian context, a complete solution hasbeen reached in the Lorentzian case. As a consequence, in [1] the authors defined adifferent type of Osserman conditions: the null Osserman conditions with respect to aunit timelike vector tangent to a Lorentz manifold (see also [2]).
It is natural to study the null Osserman conditions in a Lorentz almost contact man-ifold and, more generally, in a Lorentz g. f . f -manifold, where one finds that none ofthe above types of Osserman conditions can be satisfied.
This motivated us to introduce another kind of null Osserman condition, adaptedto the case of Lorentz g. f . f -structures, that we called ϕ-null Ossermann condition,which the present talk deals with. We study the links between the ϕ-null Ossermancondition and the behaviour of the ϕ-sectional curvature, with a particular attentiontowards the relationships between the constancy of the ϕ-sectional curvature andthe ϕ-null Osserman condition in a Lorentz S-manifold. We state an algebraic char-acterization of ϕ-null Osserman Lorentz S-manifolds with two characteristic vectorfields, finally giving some examples of such manifolds.
References
[1] E. García-Río, D.N. Kupeli and R. Vázquez-Abal, On a problem of Osserman inLorentzian geometry, Differential Geom. Appl. 7 (1997), 85–100.
[2] E. García-Río, D.N. Kupeli and R. Vázquez-Lorenzo, Osserman manifolds in
semi-Riemannian geometry, Lecture Notes in Mathematics, 1777, Springer,Berlin, 2002.
[3] R. Osserman, Curvature in the eighties, Amer. Math. Monthly 97 (1990), 731–756.
43
Semi-Riemannian submersions and ϕ-null Osserman con-ditions on Lorentzian S-manifolds
Angelo V. [email protected]à degli studi di Bari
Letizia [email protected]à degli studi di Bari
We present some results obtained while studying certain properties of the Jacobi cur-vature operator ([3]), which are related with a new condition of Osserman-type (ϕ-null Osserman condition), recently introduced in [3] in a context involving Lorentziang. f . f -manifolds, together with semi-Riemannian submersions.
Namely, on a Lorentz S-manifold (M, ϕ, ξα, ηα, g), we first find a relationship be-tween the ϕ-null Osserman condition and the classical Osserman condition ([3]),with respect to a unit tangent vector in Im(ϕ) at a point p ∈ M. Then we use thisresult to examinate some properties of projectabi-lity involving the ϕ-null Osser-man condition in (toroidal) principal bundles with ϕ-null Osserman Lorentzian S-manifolds as total space, and base space which can be either a Lorentzian Sasakianmanifold or a Kählerian manifold (see also [1]). Some examples are given at theend.
References
[1] L. Brunetti, A.M. Pastore, Some results on indefinite globally framed f -structures and toroidal principal bundles, preprint, 2010.
[2] L. Brunetti, An Osserman-type condition on Lorentz S-manifolds, preprint,2011.
[3] E. García-Río, D.N. Kupeli, R. Vázquez-Lorenzo, Osserman Manifolds in
Semi-Riemannian Geometry, Lecture Notes in Mathematics, 1777, Springer,Berlin, 2002.
44
The existence of homogeneous geodesics in homogeneouspseudo-Riemannian and affine manifolds
Zdenek Dušek ([email protected])Palacky University
It is well known that any homogeneous Riemannian manifold admits at least onehomogeneous geodesic through each point. For the pseudo-Riemannian case, even ifwe assume reductivity, this existence problem was open. We shall use a purely affinemethod to the existence problem. As the main result we prove that any homogeneousaffine manifold admits at least one homogeneous geodesic through each point. Asan immediate corollary, we have the same result for the subclass of all homogeneouspseudo-Riemannian manifolds.
References
[1] Z. Dušek, The existence of homogeneous geodesics in homogeneous pseudo-Riemannianand affine manifolds, J. Geom. Phys 60 (2010), 687–689.
45
Extrinsically flat surfaces of space forms and the geometricstructure on the space of oriented geodesics
Atsufumi Honda ([email protected])Tokyo Institute of Technology
It is well known that any complete extrinsically flat surface in the 3-sphere S3 must betotally geodesic, where “extrinsically flat” means vanishing of the Gauss-Kroneckercurvature. However, if one admits some singularities, there are many non-trivialcomplete extrinsically flat surfaces in S3. In this talk, we shall introduce a representa-tion formula for extrinsically flat surfaces in S3 in terms of a pair of two curves in the2-sphere S2.
The main idea for the representation formula is the correspondence between ruledsurfaces in S3 and curves in L(S3), where L(S3) is the space of oriented geodesicsin S3. Then we use the metric on L(S3) associated to the minitwistor complex struc-ture.
We shall also introduce some related results for anti-de Sitter 3-space H31 .
References
[1] N. J. Hitchin, Monopoles and geodesics, Comm. Math. Phys. 83 (1982), 579–602.[2] A. Honda, Isometric Immersions of the Hyperbolic Plane into the Hyperbolic
Space, preprint, 2010, arXiv:1009.3994.
46
Non-null Helicoidal Surfaces as Non-null Bonnet Surface
Abdullah [email protected] Coruh University
Soley [email protected] University
In this paper, non-null helicoidal surfaces defined by [8], [9] and timelike Bonnetsurfaces classified by [10] are taken into consideration and it is shown that the non-null helicoidal surfaces in Loretzian space forms which admit one-parameter familyof isometric deformation preserving the mean curvature are non-null bonnet sur-faces.
References
[1] A. Fujioka, J. Inoguchi, Timelike Bonnet surfaces in Lorentz space forms, Differ-ential Geom. Appl., 18 (2003), 103–111.
[2] A. Fujioka, J. Inoguchi, Spacelike surfaces with harmonic inverse mean curva-ture, J. Math. Sci. Univ. Tokyo, 7 (2000), 657–698.
[3] B. O’Neill, Semi-Riemannian geometry, Academic Press, New York, 1983.[4] J. Park, Lorentzian surfaces with constant curvatures and transformations in the
3-dimensional Lorentzian space, J. Korean Math. Soc., 45 (2008), 41–61.[5] H. S. Kim, Y. Kim, A certain complete space-like hypersurface in Lorentz man-
ifolds, Balkan J. Geom. Appl., 9 (2004), 32–43,[6] I. M. Roussos, The helicoidal surfaces as Bonnet surfaces, Tohoku Math. J., 40
(1988), 485–490.[7] L. J. Alias, A. Ferrandez, P. Lucas, Surfaces in the 3-dimensional Lorentz-
Minkowski space satisfying ∆x = Ax + B, Pacific J. Math., 156 (1992), 201–208.[8] P. Ira, J. A. Pastor, Helicoidal maximal surfaces in Lorentz-Minlowski space,
Monatsh. Math., 140 (2003), 315–334.[9] R. Lopez, E. Demir, Helicoidal surfaces in Minkowski space with constant mean
curvature and constant Gauss curvature, 2010, arXiv:1006.2345v2.[10] W. Chen, H. Li, On the classification of the timelike Bonnet surfaces, in: Geometry and
Topology of Submanifolds, 10, Chern, S. Chen, W., Shelton Street, Covent Garden,London, 18–31.
[11] W. Chen, H. Li, Bonnet surfaces an isothermic surfaces, Result. Math., 31 (1997),40–52.
47
Singularities of the asymptotic completion of developableMöbius strips
Kosuke Naokawa ([email protected])Tokyo Institute of Technology
Let U be an open domain in Euclidean two-space R2 and f : U −→ R3 a C∞ map. Apoint p ∈ U is called a singular point of f if the Jacobi matrix of f is of rank less than2 at p. Let
F(s, u) = γ(s) + uξ(s) (|u| < ε)
be a ruled Möbius strip immersed in R3, where ε > 0, γ(s) is a generating curve andξ(s) is a ruling vector field of F. Then, the C∞ map
F(s, u) = γ(s) + uξ(s) (u ∈ R)
is called the asymptotic completion (or a-completion) of the immersed strip F. It is well-known that complete and flat (i.e. zero Gaussian curvature) surfaces immersed inR3 are cylindrical. This fact implies that the a-completion of a developable Möbiusstrip (i.e. a flat ruled Möbius strip) must have singular points. Since the most genericsingular points appeared on developable surfaces are cuspidal edge singularities(cf. [1]), we are interested in how often singular points other than cuspidal edgesingularities appear on the a-completion of a developable Möbius strip. We have thefollowing result:
Proposition. The asymptotic completion of a developable Möbius strip has at least onesingular point other than cuspidal edge singularities.
A developable Möbius strip which contains a closed geodesic is called a rectifyingMöbius strip. Roughly speaking, a rectifying strip can be constructed from an iso-metric deformation of a rectangular domain on a plane. We also prove the followingassertion:
Theorem. The asymptotic completion of a rectifying Möbius strip has at least three singularpoints other than cuspidal edge singularities.
These lower bounds of the numbers of non-cuspidal-edge singularities in the propo-sition and the theorem are both sharp.
References
[1] S. Murata and M. Umehara, Flat surfaces with singularities in Euclidean 3-space, J.Diff. Geom. 82 (2009), 279–316.
48
Second-order Lagrangians admitting a first-orderHamiltonian formalism
M. Eugenia Rosado ([email protected])Universidad Politécnica de Madrid
Let p : E→ N be an arbitrary fibred manifold over a connected n-dimensional man-ifold N oriented by a volume form v = dx1 ∧ · · · ∧ dxn, and let pk : JkE → Nbe the bundle of k-jets of local sections of p, with projections pk
l : JkE → Jl E forevery k ≥ l. Every fibred coordinate system (xj, yα) on E for the projection p,1 ≤ j ≤ n, 1 ≤ α ≤ m = dim E − dim N, induces a coordinate system (xj, yα
I ),on the r-jet bundle, where I = (i1, . . . , in) ∈ Nn is an integer multi-index of order|I| = i1 + · · ·+ in ≤ r; namely,
yαI (jrxs) =
∂|I|(yα s)∂(x1)i1 · · · ∂(xn)in
(x),
where s is a local section of p defined on a neighbourhood of x ∈ N. We use the
notations I = (j) = (0, . . . , 0,(j1, 0, . . . , 0) ∈Nn and yα
(j) = yαj .
The Legendre form of a second-order Lagrangian density Λ = Lv defined on p : E→N, where L ∈ C∞(J2E), is the V∗(p1)-valued p3-horizontal (n− 1)-form ωΛ on J3Eis locally given by (e.g., see [3, 5]),
ωΛ = (−1)i−1Li0α vi ⊗ dyα + (−1)i−1Li(j)
α vi ⊗ dyαj ,
where vi = dx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn, and
Li(j)α = 1
2−δij
∂L∂yα
(ij), (1)
Li0α =
∂L∂yα
i− 1
2−δijDj
(∂L
∂yα(ij)
), (2)
where Dj denotes the “total derivative ”with respect to the coordinate xj, i.e.,
Dj =∂
∂xj +∞
∑|I|=0
m
∑α=1
yαI+(j)
∂
∂yαI
.
The Poincaré-Cartan form attached to Λ is then defined to be the ordinary n-formon J3E given by (e.g., see [3], [5]),
ΘΛ = (p32)∗θ2 ∧ωΛ + Λ,
49
where θ2 is the second-order structure form on J2E locally given in coordinates asfollows (cf. [2], [4]):
θ2 =(
dyα − yαi dxi
)⊗ ∂
∂yα+(
dyαh − yα
(hi)dxi)⊗ ∂
∂yαh
,
and the exterior product of (p32)∗θ2 and the Legendre form, is taken with respect to
the pairing induced by duality, V(p1)×J1E V∗(p1)→ R.
The most outstanding difference with a first-order Lagrangian density is that theLegendre and Poincaré-Cartan forms associated with a second-order Lagrangiandensity are generally defined on J3E, thus increasing by one the order of the La-grangian density Λ.
For certain second-order Lagrangian densities it is well known that the Poincaré-Cartan form is projectable onto J2E; e.g., see [1]. More precisely, the Poincaré-Cartanform of a second-order Lagrangian projects onto J2E if and only if the followingsystem of PDEs holds (cf. [1]):
12−δib
∂2L
∂yβac∂yα
ib
+ 12−δia
∂2L
∂yβbc∂yα
ia
+ 12−δic
∂2L
∂yβab∂yα
ic
= 0,
for all indices 1 ≤ a ≤ b ≤ c ≤ n, α, β = 1, . . . , m.
More surprisingly, there exist second-order Lagrangians for which the associatedPoincaré-Cartan form projects not only on J2E but also on J1E. Notably, this is thecase of the Einstein-Hilbert Lagrange in General Relativity.
In this talk we obtain a characterization of such Lagrangians and we study its Hamil-tonian formalism.
References
[1] Pedro L. García, J. Muñoz Masqué, Le problème de la régularité dans le calcul desvariations du second ordre, C.R. Acad. Sci. Paris 301 Série I (1985), 639–642.
[2] J. Muñoz Masqué, Formes de structure et transformations infinitésimales de contactd’ordre supérieur, C.R. Acad. Sci. Paris 298, Série I (1984), 185–188.
[3] —, An axiomatic characterization of the Poincaré-Cartan form for second-order varia-tional problems, Lecture Notes in Math. 1139, Springer-Verlag 1985, pp. 74–84.
[4] D. J. Saunders, The Geometry of Jet Bundles, Cambridge University Press, Cam-bridge, UK, 1989.
[5] D. J. Saunders, M. Crampin, On the Legendre map in higher-order field theories,J. Phys. A: Math. Gen. 23 (1990), 3169–3182.
50
Biharmonic submanifolds in Euclidean spheres
Cezar [email protected]. I. Cuza University ofLasi
Adina [email protected]. I. Cuza University ofLasi
Stefano [email protected]à degli Studi diCagliari
Biharmonic maps between Riemannian manifolds represent a natural generalizationof harmonic maps. In particular, a Riemannian immersion into a Euclidean space isa biharmonic map if and only if it is biharmonic in the sense of Chen. In this talk weshall survey old and new results on biharmonic submanifolds in Euclidean spheres.We shall present classification results for biharmonic hypersurfaces (according to thenumber of distinct principal curvatures) and biharmonic submanifolds with parallelmean curvature vector field.
Hypersurfaces in non-flat Lorentzian space forms satisfy-ing Lkψ = Aψ + b
H. Fabián Ramí[email protected] de Murcia
Pascual [email protected] de Murcia
We study hypersurfaces either in the De Sitter space Sn+11 ⊂ Rn+2
1 or in the anti DeSitter space Hn+1
1 ⊂ Rn+22 whose position vector ψ satisfies the condition Lkψ =
Aψ + b, where Lk is the linearized operator of the (k + 1)th mean curvature of thehypersurface, for a fixed k = 0, . . . , n − 1, A ∈ R(n+2)×(n+2) is a constant matrixand b ∈ Rn+2 is a constant vector. For every k, we prove that when A is self-adjointand b = 0, the only hypersurfaces satisfying that condition are hypersurfaces withzero (k + 1)-th mean curvature and constant k-th mean curvature, open pieces ofstandard pseudo-Riemannian products in Sn+1
1(Sm
1 (r)× Sn−m(√
1− r2), Hm(−r)×Sn−m(
√1 + r2), Sm
1 (√
1− r2)× Sn−m(r), Hm(−√
r2 − 1)× Sn−m(r))
open pieces ofstandard pseudo-Riemannian products in Hn+1
1(Hm
1 (−r)× Sn−m(√
r2 − 1),Hm(−
√1 + r2)× Sn−m
1 (r), Sm1 (√
r2 − 1)×Hn−m(−r), Hm(−√
1− r2)×Hn−m(−r))
and open pieces of a quadratic hypersurfaces
x ∈ Mn+1c : 〈Rx, x〉 = d
where
R is a self-adjoint constant matrix whose minimal polynomial is t2 + at + b, witha2 − 4b ≤ 0, and Mn+1
c stands for Sn+11 ⊂ Rn+2
1 or Hn+11 ⊂ Rn+2
2 . When Hk isconstant and b is a non-zero constant vector, we show that the hypersurface is totallyumbilical, and then we also obtain a classification result.
51
An optimization problem on the Lie group SO(2, 1)
Ilgin Sager ([email protected])Izmir University of Economics
In this study, the motion planning problem for the rigid body systems is formulatedas an optimal control problem on the Lie group SO(2, 1) for timelike, lightlike andspacelike cases, where the cost function to be minimized is the integral of the curva-ture squarred. The coordinate free Maximum Principle is then applied to solve thisproblem. The emphasis of this study is placed on an integrable case where the nec-essary conditions for optimality can be analytically. In addition the correspondingoptimal motions are expressed in a coordinate free manner, that is they are describedcompletely in terms of the geometrically invariant natural curvatures. These optimalmotions are shown to trace helical paths which could be useful in motion interpo-lation schemes. This problem formulation is both practical for the path planningapplication considered and illuminates how the general theory of optimal control,framed curves and left- invariant Hamiltonian systems applies to this particular set-ting.
52
A Lorentz metric on the manifold of positive definite(2× 2)-matrices and foliations by ellipses
Marcos Salvai ([email protected])FaMAF-CIEM, Córdoba, Argentina
Let E be the set of all ellipses in the plane centered at zero (with axes not neces-sarily parallel to the coordinate axes), which may be canonically identified with themanifold S+ of all positive definite (2× 2)-matrices.
The group G = Gl+2 (R) acts smoothly on S+ by g · A = gAgT . Consider on G thebi-invariant metric of signature (2, 2) given at the identity by the opposite of thecanonical inner product of the split quaternions, that is, such that 〈X, X〉 = −det Xfor all X ∈ M2 (R), and endow E ∼= S+ with the Lorentz metric pushed down fromG via the canonical projection G → S+.
We use this Lorentz metric on E to describe all foliations of (open sets of) the pointedplane R2 − 0 by ellipses. More precisely, we prove that a smooth curve γ in Edetermines a foliation of an open set of the pointed plane if and only if γ is time-like. Moreover, if the curve γ in E is defined on the whole real line, 〈γ, γ〉 = −1, and〈γ, X γ〉 is a bounded function, then the corresponding foliation covers the wholepointed plane (here X denotes the unique unit future directed G-invariant vectorfield X on E ). We also provide examples of causal curves of pairwise disjoint ellipseswhich do not determine a foliation.
The general setting is the characterization of the foliations of a smooth manifold bysubmanifolds congruent to a given one by the action of a group H, in terms of the H-invariant geometry of this set of submanifolds: Let N be a smooth manifold acted onsmoothly by a group H, let M be a submanifold of N and E the set of submanifoldsof N congruent to M via G. The problem consists in describing geometrically whichsubsets F of E determine foliations of (open subsets of) N. The paradigm is the paper[1], where fibrations of S3 by great circles are characterized in this way. See also [2] (apartial generalization of [1]) and [3], with the global foliations of R3 by lines, whichincludes a pseudo-Riemannian reformulation of the main result of [1]. In our case,N is the pointed plane, M is the circle, H = G, E = E and F the set of trajectories oftime-like curves.
References
[1] H. Gluck, F. Warner, Great circle fibrations of the three-sphere, Duke Math. J. 50
(1983), 107-132.[2] M. Salvai, Affine maximal torus fibrations of a compact Lie group, International J.
Math. 13 (3) (2002) 217-226.
53
[3] M. Salvai, Global smooth fibrations of R3 by oriented lines, Bull. London Math. Soc.41 (2009) 155-163.
Clifford Cohomology of hermitian manifolds
J. [email protected] Santiago de Compostela
L. M. [email protected] Santiago de Compostela
A. M. [email protected] de Valencia
One of the fundamental objects in the study of a smooth manifold M is its bundleof exterior differential forms, Λ∗M. This is a bundle of algebras over M generatedat each point by the cotangent space and in which there is defined a natural firstorder operator, the exterior differential d. The corresponding fundamental object inthe study of a riemannian manifold is its Clifford bundle Cl(M). This is again abundle of algebras generated by the tangent space at each point, equipped with itsinner product, and in which there is another intrinsic first order operator, the Diracoperator D.
In ([1]) Michelsohn uses the Clifford multiplication to elaborate a detailed analysisof Kähler manifolds. For this she considers the bundle ClC(M) = Cl(M) ⊕ C anda triple of parallel operators L, L and H defined on it and which carry an intrinsicsl(2)-structure of ClC(M). This, together with J, yields a decomposition
ClC(M) ≡ ⊕|p+q|≤nClp,q(M).
Taking the hermitian analogues of the Dirac operator D, she obtains operators D yD such that D2 = D
2= 0, D+D = 1/2D and D is the formal adjoint of D. The
elements in Clp,q, considered as forms, are generally of mixed degrees and underthis assumption the operator D corresponds simply to ∂ + ∂∗.
In ([3]) the authors define a formally holomorphic connection over those hermitianmanifolds which satisfy the third curvature condition. The expresion for this con-nection is
∇X = ∇L.C.X − 1
2J(∇L.C.
X J)
where ∇L.C. represents the Levi-Civita connection and X ∈ TC M.
54
In this contribution we use the algebraic theory of the Clifford algebra ClC(M) de-veloped by Michelsohn and this formally holomorphic connection to obtain similar
operators to D and D, D∇ and D∇ respectively, on certain hermitian non Kähler
manifolds, and which satisfy similar properties as, for example, (D∇)2 = (D∇)2 = 0
and D∇ is the formal adjoint of D∇.
References
[1] Bismut, J-M. A local index theorem for non Kähler manifolds. Math. Ann., 284,(1989), 681–699.
[2] Gray, A., Barros, M., Naveira, A. M., Vanhecke, L. The Chern numbers of holo-morphic vector bundles and formally holomorphic connections of complex vec-tor bundles over almost complex manifolds. J. Reine Angew. Math. 314, (1980),84–98
[3] Michelsohn, M. L. Clifford cohomology of Kähler manifolds. American Journalof Mathematics, 102, No. 6, (1980), 1083–1146
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Decoupling and Exact Solutions of Einstein Equations inAlmost Kähler Variables
Sergiu I. Vacaru ([email protected])University “Al. I. Cuza” Iaşi
We prove that the Einstein equations written in almost Kähler variables have a de-coupling property which allows us to construct solutions in very general forms fol-lowing methods elaborated in Refs. [1]. Such generic off-diagonal metrics are deter-mined by corresponding classes of generating and integration functions depending,in general, on all spacetime coordinates. The almost Kähler variables are importantfor performing deformation and A-brane quantization of Einstein gravity and elabo-rating noncommutative generalizations [2] and constructing quantum corrections tosolutions. We study geometric criteria when generic off-diagonal solutions a) defineLorenz manifolds and satisfy the Cauchy problem; b) generate solitonic hierarchiesand model geometric Ricci flows of low dimensional geometries on four dimensionalspacetime manifolds. There are considered extensions of the method for constructingexact solutions in modified theories of gravity (for instance, with extra dimensions,with scaling anisotropy and/or generalized Lagrange–Finsler spaces). Finally, weprovide examples and speculate on new classes of physically important solutionsand discuss geometric methods of their quantization.
1. S. Vacaru, Int. J. Geom. Meth. Mod. Phys. 8 (2011) 9, arXiv: 0909.3949v1;Int. J. Theor. Phys. 49 (2010) 2753, arXiv: 1003.0043v1; 4 (2007) 1285, arXiv:0704.3986
2. S. Vacaru, Int. J. Geom. Meth. Mod. Phys. 6 (2009) 873, arXiv: 0810.4692; J.Geom. Phys. 60 (2010) 1289, arXiv: 0709.3609; Acta Applicandae Mathemat-icae 110 (2010) 73, arXiv: 0810.0707; J. Math. Phys. 50 (2009) 073503, arXiv:0806.3814; Class. Quant. Grav. 27 (2010) 105003, arXiv: 0907.4278
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Socialevents
Social events
Tuesday, September 6
08:15 Registration
08:45 Opening Ceremony
Aula Magna, Facultad de Ciencias.
20:30 Rectorat university cocktail reception atHospital Real building offered by the University of Granada.
Wednesday, September 7
21:00 Social dinner at the “Carmen de los Chapiteles”,Camino de la Fuente del Avellano, no
4.
Thursday, September 8
Afternoon Guided visit to the Alhambra monument.Information and tickets sold by Eurocongres. The visit includes guide and busservice from and back to the Faculty of Sciences.Bus departure at 16:15 at the front door of the Science Faculty.
Friday, September 9
18:00 Closing Ceremony
Aula Magna, Facultad de Ciencias.
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Aula MagnaFacultad de CienciasUniversidad de GranadaFuentenueva s/n18071 Granada
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Avda. de la Constitución no18-bq. 4
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