Mathematics

Embed Size (px)

DESCRIPTION

:)

Citation preview

  • Determining a Metric by Boundary

    Single-Time Flow Energy

    ARIANA PITEAFaculty of Applied Sciences,Department of Mathematics I,

    University Politehnica of Bucharest,Splaiul Independentei 313

    RO-060042, Bucharest, [email protected]

    Abstract: Our theory of determining a tensor by single-time flow energy is similar to those developed bySharafutdinov. Section 1 refines the theory of potential curves determined by a flow and a Riemannianmetric. Section 2 defines the boundary energy of a potential curve and proves that the problem ofdetermining a metric from a single-time flow boundary energy cannot have a unique solution. Section 3linearizes the above-mentioned problem and defines the notion of single-ray transform.

    Key Words: potential curves, least squares Lagrangian, single-ray transform, boundary energy.

    1 Potential curves

    Let (M, g) be a compact Riemannian mani-fold with the boundary M and of dimension n.We consider x = (xi) local coordinates on themanifold (M, g), (Gjk,`) and (Gijk) its Christof-fel symbols of the first and of the second type,respectively.

    Let : [0, 1] M , (t) = x, x(t) =(x1(t), . . . , xn(t)) be a C-curve. We want to ap-

    proximate the velocitydx

    dtof components

    dxi

    dtby

    a C-distinguished vector field X of componentsXi(x), in the sense of least squares. For that we

    build the flowdxi

    dt(t) = Xi(x(t)), 1, n, x(0) = p,

    x(1) = q, where p and q are two points from theboundary M of the manifold M , and the leastsquares Lagrangian (flow energy density),

    L(t, x(t), x(t)) =12gij(x(t))[xi(t)Xi(x(t))]

    [xj(t)Xj(x(t)],

    where xi(t) =dxi

    dt(t).

    The geometric dynamics (ODEs or PDEs) isa Lagrangian dynamics (ODEs or PDEs) deter-mined by a least squares Lagrangian attached toa (single-time or multi-time) flow and a pair ofRiemannian metrics, one in the source space andother in the target space.

    Let us look for the Euler-Lagrange prolonga-tion of the flow obtained as Euler-Lagrange ODEsproduced by L. The extremals of L are called po-tential curves. These curves are geodesics [4].

    Theorem 1.1 The extremals of L are de-scribed by the ODEs:

    dtxi = gi`gkj(`Xk)Xj + F ij xj , i = 1, n

    x(0) = p, x(1) = q,

    where:

    dtxi = xi +Gijk x

    j xk, i = 1, n,

    F ij = jXi gi`gkj`Xk, i, j = 1, n; (1)

    jXi = Xi

    xj+GijkX

    k, i, j = 1, n. (2)

    Proof. We computeL

    xk=

    12gijxk

    xi xj gijxk

    xiXj +12gijxk

    XiXj

    gij xj Xi

    xk+ gij

    Xi

    xkXj ;

    L

    xk= gikxi gikXi;

    d

    dt

    (L

    xk

    )=

    gikx`

    x`xi + gikxi gikx`

    xiX`

    gik(X i

    t+

    X i

    x`x`).

    WSEAS TRANSACTIONS on MATHEMATICS Ariana Pitea

    ISSN: 1109-276940

    Issue 1, Volume 7, January 2008

  • We take into account formula (1) and

    gijxk

    = G`kig`j +G`kjg`i, i, j, k = 1, n.

    If we replace these relations in Euler-Lagrangeequations,

    L

    xk d

    dt

    (L

    xk

    )= 0, k = 1, n,

    we find

    gkjxj

    dt= gij(kXi)Xj + gkj(`Xj)x`

    gij xikXj , k = 1, n.

    Transvecting by g`k and using formula (2), weobtain

    xi

    dt= gikg`j(kX`)Xj + F ij xj , i = 1, n

    Definition 1.1 The map C([0, 1],M),(t) = x, which verifies the ODEs from the pre-vious theorem is called potential curve associatedto the d-vector field X.

    Definition 1.2 The Riemannian metric g iscalled simple if there is a unique potential curvex: [0, 1]M , x(0) = p, x(1) = q, p, q M .

    2 Determining a metric byboundary flow energy

    Starting from the boundary flow energy, westudy the recovering of a tensor from the centeredmoments which determine the metric g.

    Let g be a simple metric and x: [0, 1]M thecorresponding potential curve, x(0) = p, x(1) = q,p, q M .

    Definition 2.1 Let p and q be two points onthe boundary M of the manifoldM .The functionEg: M M R, (p, q) 7 Eg(p, q),

    Eg(p, q) =12

    10gij(x(t))[xi(t)Xi(x(t))][xj(t)

    Xj(x(t))] dt

    is called the boundary energy of the flow along thepotential curve x: [0, 1] M ,x(0) = p, x(1) = q,p, q M .

    Open problem 1. Given an energy E, isthere a metric g that realizes this energy? Howcan these metrics be found?

    Let us show that the existence problem of themetrics with the property that E: MMRrepresents the boundary energy cannot have aunique solution.

    Let :M M be a diffeomorphism with theproperty |M = id. This diffeomorphism trans-forms the simple metric g0 into a simple metricg1 = g0, because we have

    g1(x)(, ) = g0((dx), (dx))(x),

    where dx:TxM T(x)M is the differential of.

    It can be noticed that

    X i=

    xi

    xiXi,

    whereXirepresents the distinguished vector field

    X with respect to x = (x).The metrics g0 and g1 give different families of

    potential curves with the same boundary energyE.

    Open problem 2. The problem of recover-ing a metric by boundary energy can be changedin the following way. Let g0 and g1 be simplemetrics on M . Does the equality Eg0 = Eg1 im-pliy the existence of a diffeomorphism :M M ,|M = id and g1 = g0?

    3 Linearization of the problemof finding a metric from theboundary energy

    Let us linearize the open problem 1. Let(g ) be a family of simple metrics which dependssmoothly on parameter (, ), > 0. Letp and q be two points of the border M of themanifold M and a = E(p, q), where E: M M R is the given boundary energy. Considerx : [0, a] M the potential curve correspondingto the pair (p, q), x = (x,i), x,i(t) = xi(t, ),i = 1, n. We denote g = (gij).

    The energy of deformation x is

    Eg (p, q) =12

    a0gij(x

    (t))[xi(t, )Xi(x(t, ), )]

    [xj(t, )Xj(x(t, ), )] dt.

    WSEAS TRANSACTIONS on MATHEMATICS Ariana Pitea

    ISSN: 1109-276941

    Issue 1, Volume 7, January 2008

  • Differentiating with respect to we obtain

    Eg (p, q)=

    a0

    {gij

    (x (t))[xi(t, )Xi(x (t), )]

    [xj(t, )Xj(x (t), ) + gij

    xk(x (t))

    [1

    2xi(t, )xj(t, )]

    xi(t, )Xj(x (t), ) + 12Xi(x (t), )

    Xj(x (t), )

    ]xk

    (t, ) + gij(x

    (t))

    [xi

    (t, )xj(t, )

    xi

    (t, )Xj(x (t), ) xi(t, )X

    j

    (x (t), )

    xi(t, )Xj

    xk(x (t), )

    xk

    (t, ) +Xi(x (t), )

    Xj

    (x (t), ) +Xi(x (t), )

    Xj

    xk(x (t), )

    xk

    (t, )

    ]}dt =

    a0

    gij

    (x(t, ))

    [xi(t, )xj(t, ))

    2xi(t, )Xj(x (t), +Xi(x (t), )Xj(x (t), )]dt

    +

    a0

    {gijxk

    (x (t))

    [1

    2xi(t, )xj(t, )

    xi(t, )Xj(x (t), ) + 12Xi(x (t), )Xj(x (t), )

    ]gij(x (t))[xi(t, )Xi(x (t), )]

    Xj

    xk(x (t), )

    }xk

    (t, ) dt+

    a0

    gij(x (t))[xj(t, )

    Xj(x (t), )]xi

    (t, ) dt

    a0

    gij(x (t))[xi(t, )

    Xi(x (t), )]Xj

    xk(x (t), )

    }xk

    (t, ) dt+

    a0

    gij(x (t))

    [xi(t, )Xi(x (t), )]Xj

    (x (t), ) dt. (3)

    Integrating by parts, then considering = 0and using the fact that

    xi

    a0

    = 0,

    the third integral from relation (3) becomes

    I3= a0

    d

    dt

    =0

    gij(x (t))[xj(t, )Xj(x (t),)]x

    i

    (t, 0) dt

    = a0

    {g0ijx`

    (x0(t))x`(t, 0)[xj(t,0)Xj(x0(t), 0)]

    +g0ij(x0(t))

    [xj(t, 0)X

    j

    x`(x0(t), 0)

    ]x`(t,0)

    }xi

    (t,0) dt.

    Relation (3) becomes, at = 0:

    =0

    Eg (p, q) =

    a0

    fij(x0(t))[xi(t, 0)xj(t, 0)

    2xi(t, 0)Xj(x0(t), 0) +Xi(x0(t), 0)Xj(x0(t), 0)] dt

    +

    a0

    {g0ijxk

    (x0(t))

    [1

    2xi(t, 0)xj(t, 0)

    xi(t, 0)Xj(x0(t), 0) + 12Xi(x0(t), 0)Xj(t, x0(t), 0)

    ]

    g0ij(x0(t))[xi(t, 0)Xi(x0(t), 0)]Xj

    xk(x0(t), 0)

    }xk

    (t, 0) dt+

    { a0

    g0ijx`

    (x0(t))x`(t, 0)[xj(t, 0)

    Xj(x0(t), 0)]+g0ij(x0(t))[xj(t, 0) X

    j

    x`(x0(t), 0)

    x`(t, 0)

    ]xi

    (t, 0) dt

    } a0

    g0ij(x0(t))[xi(t, 0)

    Xi(x0(t), 0)]Xj

    (x0(t), 0) dt, (4)

    where fij =12

    =0

    gij .

    Making the sum of the second and third inte-grals of (4), we obtain

    I2 + I3 = 2g0kj

    xi(x0(t))xi(t, 0)xj(t, 0) +

    g0ijxk

    (t, 0)

    [2xi(t, 0)Xj(x0(t), 0) +Xi(x0(t), 0)Xj(x0(t), 0)]

    2g0ij(x0(t))[xi(t, 0)

    Xj

    xk(x0(t), 0) X

    i

    xk(x0(t), 0)

    Xj(x0(t), 0)

    ]+ 2

    g0kjx`

    (x0(t))x`(t, 0)Xj(x0(t), 0)

    2g0kj(x0(t))[xj(t, 0) X

    j

    x`(x0(t), 0)x`(t, 0)

    ]}xk

    (t, 0)dt

    =1

    2

    a0

    {2g0`k(x0(t))G`ij(x0(t), 0)xi(t, 0)xj(t, 0)

    +[Gki,j(x0(t), 0)+Gkj,i(x

    0(t), 0)][2xi(t, 0)Xj(x0(t), 0)

    +Xi(x0(t), 0)Xj(t, x0(t), 0)]2g0ij(x0(t))[xi(t, 0)

    Xj

    xk(x0(t), 0) X

    i

    xk(x0(t), 0)Xj(x0(t), 0)

    ]+2[Gk`,j(x

    0(t), 0) +G`j,k(x0(t), 0)]x`(t, 0)Xj(x0(t), 0)

    2g0kj(x0(t))[xj(t, 0) X

    j

    t(x0(t), 0) X

    j

    x`(x0(t), 0)

    WSEAS TRANSACTIONS on MATHEMATICS Ariana Pitea

    ISSN: 1109-276942

    Issue 1, Volume 7, January 2008

  • x`(t, 0)

    ]}xk

    (t, 0)dt=

    1

    2

    a0

    {2g0`k(x0(t))G`ij(x0(t), 0)

    xi(t, 0)xj(t, 0) 2g0iq(x0(t))xi(t, 0)kXq(x0(t), 0)

    +2g0jq(x0(t))

    [Gqk`(x

    0(t), 0)X`(x0(t), 0)Xj(x0(t), 0)

    +Xq

    xk(x0(t), 0)Xj(x0(t), 0)

    ]+ 2g0qk(x

    0(t))

    Gqj`(x0(t), 0)x`Xj(x0(t), 0) + 2g0kj(x

    0(t))

    Xj

    x`(x0(t), 0)x`(t, 0)

    }xk

    (t, 0) dt

    =

    a0

    { g0jk(x0(t))[xj(t, 0) g0iq(x0(t))

    xi(t, 0)kXq(x0(t), 0) + g0qk(x0(t))Gqj`(x0(t), 0)

    x`(t, 0)Xj(x0(t), 0) + g0kj(x0(t))

    Xj

    x`(x0(t), 0)

    x`(t, 0)

    }xk

    (t, 0) dt =

    a0

    [ g0`j(x0(t))

    (kX`(x0(t), 0)Xj(x0(t), 0)g0ki(x0(t))F ij (x0(t), 0)

    xj(t, 0) g0iq(x0(t))xi(t, 0)kXq(x0(t), 0)

    +g0jq(t)kXq(x0(t), 0)Xj(x0(t), 0)

    +g0qk(x0(t))Gqj`(x

    0(t), 0)x`(t, 0)

    Xj(x0(t), 0)

    ]xk

    (t, 0) dt =

    a0

    [ g0ki(x0(t))

    F ij (x0(t), 0)xj(t, 0) g0iq(x0(t))xi(t, 0)

    (kXq)(x0(t), 0) + g0qk(x0(t))Gqj`(x0(t), 0)x`(t, 0)

    Xj(x0(t), 0) + g0kj(x0(t))

    Xj

    x`(x0(t), 0)

    x`(t, 0)

    ]xk

    (t, 0) dt =

    a0

    xj(t, 0)

    { g0ki(x0(t))

    [(jXi)(x0(t), 0) giq0 (x0(t))g0`j(x0(t))

    (qX`(x0(t), 0))] g0jq(x0(t))(kXq)(x0(t), 0)

    +g0qk(x0(t))Gq`j(x

    0(t), 0)X`(x0(t), 0)

    +g0k`(x0(t))

    X`

    xj(x0(t), 0)

    }xk

    (t, 0) dt

    =

    a0

    xj(t, 0)

    [ g0ki(x0(t))(jXi)(x0(t), 0)

    +g0`j(x0(t))(kX`)(x0(t), 0) g0jq(x0(t))

    (kXq)(x0(t), 0)+g0qk(x0(t))Gq`j(x0(t))X`(x0(t), 0)

    +g0k`(x0(t))

    X`

    xj(x0(t), 0)

    ]xk

    (t, 0) dt = 0.

    We have obtained that

    =0

    Eg (p, q)=

    a0

    fij(x0(t))[xi(t, 0)Xi(x0(t), 0)]

    [xj(t, 0)(Xj(x0(t), 0)] dt a0

    g0ij(x0(t))[xi(t, 0)

    Xi(x0(t), 0)]Xj

    (x0(t), 0) dt,

    where fij(x0(t)) =12

    =0

    gij .

    Considering thatXj

    (x0(t), 0) = 0, and us-

    ing the functional

    If (x0) =

    a0

    fij(x0(t))[xi(t, 0)Xi(x0(t), 0)]

    [xj(t, 0)Xj(x0(t), 0)]dt

    the previous relation becomes

    =0

    Eg (p, q) = If (x0), (5)

    where x0 is the potential curve corresponding tothe points p, q from the border M of the mani-fold M .

    The function If is called the single-ray trans-form of the tensor field (fij).

    The existence of solutions of the open problem1 for the family (g ) implies the existence of a oneparameter group of diffeomorphisms (x) suchthat g = ( )g0. Explicitly

    gij = (g0k` )

    xk

    xix`

    xj, (6)

    where (x)=(1(x, ), . . . , n(x, )), x= (x).

    Theorem 3.1 Let vk(x)=

    =0

    (xk)(x, ),

    k = 1, n, vi = g0ijvj and vi;j be the covariant

    derivative of (vi). Then the following relationholds

    fij =12(vi;j + vj;i), i, j = 1, n. (7)

    Proof. Differentiating the relation (6) withrespect to and then considering = 0, we find

    2fij=

    =0

    gij =g0k`xm

    vmxk

    xix`

    xj

    +g0k`

    xi

    (

    =0

    xk)x`xj

    WSEAS TRANSACTIONS on MATHEMATICS Ariana Pitea

    ISSN: 1109-276943

    Issue 1, Volume 7, January 2008

  • +g0k`xk

    xi

    xj

    (

    =0

    x`)

    =g0k`xm

    vmki `j + g

    0k`

    vk

    xi`j + g

    0k`

    ki

    v`

    xj

    =g0ijxq

    vq + g0jqvq

    xi+ g0iq

    vq

    xj.

    On the other hand,

    vi;j + vj;i =vi

    xjGmij vm +

    vjxi

    Gmjivm =g0imxj

    vm

    +g0imvm

    xj+

    g0jmxi

    vm + g0jmvm

    xi

    gmq0(g0jqxi

    +g0iqxj

    g0ij

    xq

    )g0msv

    s

    =g0ijxq

    vq + g0jqvq

    xi+ g0iq

    vq

    xj,

    and relation (7) is provedTherefore, the following generalization of the

    open problem 1 appears. To what extent do theintegrals

    If (x) = a0fij(x(t))[xi(t)Xi(x(t))]

    [xj(t)Xj(x(t))]dt

    determine the tensor (fij)?

    References:

    [1] M. Neagu: Riemann-Lagrange Geometry on 1-JetSpaces, Matrix Rom, Bucharest, 2005.

    [2] A. Sharafutdinov: Integral Geometry of TensorFields, VSPBV Utrecht, 1994.

    [3] C. Udriste: Convex Functions and OptimizationMethods on Riemannian Manifolds, Kluwer Aca-demic Publishers, 1994.

    [4] C. Udriste, M. Ferrara and D. Opris: EconomicGeometry Dynamics, Geometry Balkan Press,2004.

    [5] C. Udriste, Ariana Pitea and J. Mihaila: Deter-mination of Metrics by Boundary Energy, BalkanJ. Geom. Appl., vol. 11, no. 1 (2006), pp. 131-143.

    [6] C. Udriste, Ariana Pitea and J. Mihaila: KineticPDEs System on the First Order Jet Bundle,Proc. 4-th Int. Coll. Math. Engng&Num. Phys.

    [7] C. Udriste and M. Postolache: Atlas of MagneticGeometric Dynamics, Geometry Balkan Press,2001.

    [8] H. Urakawa: Calculus of Variations and Har-monic Maps, Shokabo Publishing Co. Ltd.,Tokyo, 1990.

    WSEAS TRANSACTIONS on MATHEMATICS Ariana Pitea

    ISSN: 1109-276944

    Issue 1, Volume 7, January 2008