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CHAPTER-I
INTRODUCTION
(1.1) HISTORICAL BACKGROUND
The study of differentiable manifolds besides being an
interesting subject in itself , has become useful in an ever
increasing number of disciplines of pure as well as applied
mathematics . Differentiable manifolds constitute the base for the
study of advanced calculus and modern analysis . Newton and
Leibnitz had laid the foundation of calculus . Many results
concerning surface in 3- space were obtained by gauss in the first
half of the nineteenth century and in 1854 Riemann laid the
foundation for a more abstract approach . Later on Civita and Ricci
developed the concept of parallel translation in the classical
language of tensors.
In 1930 , Schouten and Dantzing introduced the concept of a
complex structure and a Hermitian metric in a differentiable
manifold and called it as a complex manifold . The idea of
Kaehlerian structure on a complex manifold was initially presented
by Kaehler in 1933 . Ehresmann in 1947 defined an almost
complex manifold as an even dimensional differentiable manifold
Mn (n = 2m) of differentiability class cr+1 such that there exists a
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tensor field f of type (1,1) and differentiability class cr , satisfying
the equation F2+In= 0 , In being unit tensor field . Later on , Weil in
1947 also pointed out that in a complex manifold ,the type (1 , 1)
tensor field F Satisfying F2 = - In , F is called to be an almost complex
structure to Mn . Newlander and Nirenberg in 1957 studied the case
when the complex space is merely differentiable . Nijenhuis in
1951 introduced a very important tensor named as Nijenhuis tensor
. The differential geometry of tangent and cotangent bundles was
studied by sasaki in 1958 , Sasaki in 1960 and Hsu in 1962 defined
and studied almost contact structure and its integrability conditions.
Other type of structure on differentiable manifolds were
defined and studied by Helgason in 1962 , Yano in 1963, Yano and
Davies in 1963 ,Goldberg in 1963, Duggal in 1964,Yano and
Ishihara in 1965, R.S.Mishra in1965, Yano and Kobayashi in 1966,
Yano and Patterson in 1967 , Blair in 1970 , etc.
Vanzura in 1972 defined and studied an almost r - contact
structure manifold . In 1976 , Upadhyay and Dube Studied almost
contact hyperbolic (f , g ,η , ξ )-structure .Sato in 1976 studied a
structure similar to an almost contact structure and later on this
structure was called an almost Para contact structure.
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There are two well known classes of submanifolds namely
invariant submanifold and anti-invariant submanifolds. In the first
case tangent space of the submanifold remains invariant under the
action of almost complex structure F where as in the second case ,
it is mapped into the normal space. Study of differential geometry
of CR – submanifolds is a generalization of invariant submanifold
of an almost Hermitian manifold was initated by Bejancu in 1978
and was followed by several geometers .In 1980 Pal and Mishra
studied Hypersurfaces of almost hyperbolic Hermite manifolds . In
1981 , Dube and Mishra studied hypersurface immersed in an
almost hyperbolic Hermittian manifold . Chen in 1981 further
generalized the concept of CR-submanifold that introduced generic
submanifold .
Bejancu and Papaghuic initiated the study of semi-invariant
submanifolds in a Sasakian manifolds in 1981, Kobayashi in 1981,
Yano and Kon in 1983 studied the same concept under the name
contact CR – submanifold .
A semi - invariant submanifold is nothing but the extension
of the concept of CR - submanifold of a Kaehlerian manifold to
submanifold of an almost contact metric manifold .
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In 1985 Oubina introduced a new class of almost contact
structure namely the trans Sasakian structure . In 1991 Shahid
studied CR -submanifolds of trans Sasakian manifold . In 1993
Shahid studied semi-invariant submanifolds of a nearly Sasakian
manifold .
Besides this several geometers like : K.Minoru , Oubina , K.
Matsumoto , K. Kenmotsu , Ram Nivas , Shahid , Adler , B. N.
Parsad , P.N. Pandy, K K Dubey , Jafar Ahsan , H. D. Pandy ,
H.S.Shukla , Dhruwa Naraian , Asha Srivastava , S . K . Srivastava
and other geometers provided new dimensions to the theory of
differentiable manifold .
(1.2) DIFFERENTIABLE MANIFOLD
Let be Rn the set of real number and integer n > 0. Then the
maps ui : Rn → R for ui (x1 , x2 ,……….., xn) = xi are called the
natural coordinate functions on Rn .
A map f from an open set A Ì Rn into R is called to be a
differentiability class Cr on A if it possesses continuous partial
derivatives on A of all order ≤ r upto r Î I , I being the set of
integer . If f is continuous from A into R , f is said to be of class C0
on A . While f is Cr on A for all r , we say that f is C¥ on A . Also ,
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f is real analytic on A , f is said to be of class Cω on A . A Cω
function is a C¥ function but converse is not true[24] .
Fig. 1.2.1
Let Mn be a set of n- dimensional space containing an open
set U. An n - coordinate pair (f , U) on Mn is a pair consisting of a
subset U of Mn and a one to one map f of U onto an open set in Rn
is the product space of ordered n - tuples of real numbers . An n -
coordinate pair (f , U) is called Cr (C¥) related to another n -
coordinate pair (q,V) if the mapping q o f-1 and f o q-1 are Cr (C¥)
maps . A Cr (C¥) n - subatlas on Mn is a collection of Cr (C¥)
related n - coordinate pairs (fh , Uh) such that the union of the set
q
f Ο q-1
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Uh is M . A maximal collection of Cr (C¥) related n - coordinate
pairs on Mn is called a Cr (C¥) n - atlas on M . An n-dimensional
Cr(C¥) manifold is a set Mn together with Cr(C¥) n – atlas . We
denote this manifold by Mn and call it as a Cr(C¥) n - manifold . An
n - dimensional Cr manifold for which r ¹ 0 , is called a
differentiable manifold or smooth manifold of class Cr. If r = 0 , Mn
is called a topological manifold [39].
(1.3) VECTOR , FORMS AND TENSORS
Let V1 be an m- dimensional vector space with basis {ei} of
m - linearly independent vector . The component Pi of the vector P
with respect to the basis {ei} is called a contravariant vector . Let
V1 be the dual space of V1 , then the dimensions of V1 is the same
as that of V1. Let {ei} be the basis of V1, then {ei} is a set of m -
linearly independent vector . The component B1 of the vector B of
V1 with respect to the basis {ei} is called a covariant vector .
Also , the elements of VI are covariant vectors with respect
to the basis of V1 and the elements of V1 are contravariant vectors
with respect to its own basis {ei}. But basically, the vectors of V1
are called contravariant and that of V1 are called covariant with
respect to the basis of V1. The covariant vectors are also known as
1- forms .
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Tensor are geometric object that describe linear relations
between scalars , vector , matrices and other tensors . Vector and
scalars themselves are also tensor . A tensor can be represented as a
multi-dimensional array of numerical values . The order (also
degree or rank) of a tensor is dimensionality of array needed to
represented it or equivalently , the number of indices needed to
label a component of that array. For example, a linear map can be
represented by a matrix , a 2- dimensional array and therefore is a
1- dimensional array and is a 1st – order tensor . Scalars are single
number and are thus zeroth-order tensor .
Tensor of higher order are defined by taking into account of
the tensor product of vectors . Let V2 def V1 Ä V1 , denote the
tensor product of V1 with itself and let ei Ä ei def eij . Then the
component Tij of T with respect to the basis {eij} of V2 is called
contravariant tensor of order 2 or tensor of type (2,0) . Let us
denote the tensor product of V1 with itself by V2= V1 Ä V1 and ei Ä
ej= eij . Then the component Uij of U is called covariant tensor with
respect to the basis {eij} of V2 or the covariant tensor of order 2 or
the tensor of the type (0 , 2) . Thus the tensor product of two
contravariant (or covariant) vectors is a contravariant (or covariant)
tensor of the order 2 but every contravariant (or covariant) tensor
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of the order 2 is not necessarily the tensor product of two
contravariant (or covariant) vectors.A mixed tensor of the order2 or
a tensor of the type (1,1)is defined as an element of V11
def V1Ä V1
with respect to the basis {ei} of V1.The tensor product of a
contravariant vector and a covariant vector is a mixed tensor of
order 2 but every mixed tensor of order 2 is not necessarily the
tensor product of a contravariant vector and a covariant vector .
The tensors of the type ( r , 0) and (0 , s) are defined as the tensor
product of r- contravariant vector and the tensor product of s-
covariant vectors respectively .
To define a mixed tensor of type (r , s) i.e. contravariant of
order r and covariant of order s , we take the tensor product of V1
repeated r- times and that of V1 repeated s- times and denoted it
Vsrdef V1 Ä-------------------Ä V1 Ä V1Ä-------------ÄV1 .
r-times s-times
The of mr+s vectors forms a basis of Vsr ,
where def ei1Ä---------ÄeirÄej1-----------Äejs .
The component of P are called the tensors of the
type (r , s) with respect to the basis of V1 . The tensor product of
vectors belonging of V1 and s- vectors belonging to V1 is a tensor
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of type (r , s ) but every tensor of the type (r , s) is not necessarily
the tensor product of the elements of VI repeated r – times and that
of V1 repeted s- times . The tensor of type (0,0) are known as
scalars . To verify whether a scalar valued or a vector valued
function is tensor , it is sufficient to test the linearity of the function
in all the vectors (slots) and all the 1- forms .
(1.4) TENSOR FIELDS AND TENSOR PRODUCT
A tensor field of type (r , s) is r - contravariant and s -
covariant tensor field on an open set A . It is defined as the
mapping that assigns to each point m in A tensor of type (r , s) at
that point . The set of all tensor fields of type ( r, s) on A forms a
vector space ( )ns r, m MT under usual vector addition and scalar
multiplication .
A covariant vector field w on a set A is called of class C¥ if
(i) A is open
and
(ii) w (X) is a C¥ function on A , for all C¥ contravariant
vector fields X on A .
A 0 - form on an open set A is an element of F0 (A) and a 1 -
form on A is a C¥ covariant vector field on A . A p - form on A is a
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skew - symmetric p - covariant tensor field and the set of all p -
forms on A is denoted by Fp (A) .
If a Î To, p(A) and b Î To, q(A) , then the tensor product of
covariant tensors a and b denoted by a Ä b is defined to be an
element in To, p+q (A) defind as :
(1.4.1) (a Ä b) (X1 , X2,…….. , X p+q) = a (X1 , X2….. , Xp)
b (Xp+2,….., X p+q)
For all vector fields X1 , X2 ,…..X p+q on A .
The operation of tensor product satisfies the following
relations :
(i) (a1 + a2) Ä b = a1 Ä b + a2 Ä b ,
(1.4.2) (ii) a Ä (b1 + b2) = a Ä b1 + a Ä b2
and (iii) (ba) Ä b = a Ä (bb) = b(a Ä b) ,
where b is a real number and a, a1, a2, b, b1, b2 are covariant
tensors .
However
a Ä b ¹ b Ä a in general , but (a Ä b) Ä g = a Ä (b Ä g) .
Thus tensor product a Ä b is bilinear and associative but not
symmetric in general . The tensor product of contravariant tensor
or mixed tensor can also be defined analogously.
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(1.5) TANGENT VECTOR AND TENGENT SPACE
Let Mn be a differentiable manifold of n- dimensions and b be
any point in Mn . Further F be the set of all real valued functions
that are C¥ on some neighbourhood of b . There after a vector X at
b , satisfying
(1.5.1) X in Mn , f Î F such that XF Î F ,
(1.5.2) X(f + g) = X f + X g , f , g Î F ,
(1.5.3) X (fg) = f(Xf) + g (Xf) ,
(1.5.4) and X (af) = a(Xf) , aÎR(the set of real numbers ) .
The X is called the tangent vector to Mn at b . The system
consisting of the set of all tengent vectors Tb at b , a binary
operation , say ‘ +’ satisfying
(1.5.5) X , Y Î Tb such that X+Y Î Tb ,
(1.5.6) (X+Y)f = Xf + Yf , f Î F
and an operation of scaler multiplication , satisfying
(1.5.7) f Î F , X Î Tb such that FX Î Tb ,
(1.5.8) (a X) f = a(Xf) , a ÎR (the set of real numbers)
Is a vector space called the tangent space to Vm at b denoted
by T1 .
(1.6) LIE – BRACKET
Let X and Y be two arbitrary vector fields of class C¥ of Mn,
then their Lie - bracket is a mapping
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[ ] : Mn X Mn ® Mn
defined as :
(1.6.1) [X , Y] f = X (Yf) – Y (Xf)
where f is a C¥ - functions
The Lie - bracket satisfies the following properties :
(1.6.2 a) [X , Y] (f1 + f2) = [X ,Y] f1 + [X ,Y] f2
(1.6.2 b) [X ,Y] (f1f2) = f1 [X ,Y] f2 + f2 [X ,Y] f1
(1.6.2 c) [X ,Y] + [Y, X] = 0 (skew symmetry)
(1.6.2 d) [X + Y, Z] = [X , Z] + [Y, Z] (bilinear)
and (1.6.2 e) [X , [Y, Z]] + [Y, [Z , X] ] +Z , [X , Y] = 0
(Jacobi identity) .
(1.7) CONNECTION
Let Mn be a C¥ manifold and P be any point of Mn . Further
T(p) be a tangent space to Mn at the point P and r(p)sT be a vector
space whose elements are the tensor of the type (r , s) .
A connection D is a type preserving mapping D : T(s) Å rs
rs T T ® , which assigns to each pair of C¥ vector field (X , P) where
X Î T(p) , P Î rsT , a vector field D X P such that for C¥ functions f
the following relations hold true .
(1.7.1) D X f = X f ,
(1.7.2) D X a = 0 , a Î R ,
(1.7.3) D X (Y + Z) = D XY + D X Z ,
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(1.7.4) D X (fY) = f ( D XY ) + (Xf)Y ,
(1.7.5) D X+Y Z = D X Z + D Y Z ,
(1.7.6) D fXY = f D XY ,
(1.7.7) (D X A) (Y) = X (A) (Y)) – A (D XY)
and
(1.7.8) s1r1s1r1X X)....... ,X,A,.....A( (P X X)....... ,X,A..... ,A( P) (D =
s1r1s1r1X X)....... ,X,A,.....A( P -......- X)....... ,X,A..... ,A(D P- ,
where D X is the covariant differentiation along the vector field X
with respect to the connection D , s21
X......., X , X are the vector fields
and r21
A ......., A ,A are 1- forms .
(1.8) METRIC TENSOR
In three - dimensional Euclidean space the distance ds
between the two continuous points (p , q , r) and ( p+dp , q+dq ,
r+dr ) is defined as :
ds2 = dp2 + dq2 + dr2
Then ds is called the “ element of the curve” and the axes are
rectangular Cartesian . The distance ds between two adjacent
points ii
p d p and ,......n) 2 1,(i p +=i
is determined by Riemann as follows :
2 i
ijds g ( dp dp , i , j 1, 2 , ....nj= = ) ,
where the coefficients g i j are function of pi such that :
g = | g i j | ¹ 0
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(1.9) RIEMANNIAN MANIFOLD
Let M be a n - dimensional C¥ manifold with tangent space
Tx at x Î Mn . A real valued, bilinear , symmetric and positive
definite function g on T X X , is called a Riemannian metric . An n-
dimensional C¥ manifold equipped with a Riemannian metric is
called a Riemannian manifold and the geometry of such a manifold
is termed as Riemannian geometry . The Riemannian metric g is
also called as metric tensor or fundamental tensor of type (0 , 2) .
When the conditions of positive definiteness is replaced by non –
degeneracy , then the metric is termed as indefinite or semi -
Riemannian metric .
(1.10) TORSION TENSOR
A vector valued , skew symmetric , bilinear functions T of
the type (1, 2) defined by
(1.10.1) T(X, Y) def D X Y – D YX – [X , Y] .
The tensor T is called the Torsion tensor of connection D.
The torsion tensor T satisfies the following properties :
(1.10.2) T(X ,Y) = -T(Y, X)
(1.10.3) and T(fX , hY) = fhT(X ,Y) ,
where X , Y are arbitrary vector fields and f , h are C¥-functions.
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(1.11) THE NIJENHUIS TENSOR
The Nijenhuis tensor N of a manifold Mn with an almost
complex structure F is given by[43] :
(1.11.1) N(X ,Y) = [FX , FY] – F[FX ,Y] – F[X , FY] + F2 [X ,Y]
The vanishing of the Nijenhuis tensor is the necessary and
sufficient condition for F to be integrable . Several equivalent
statements of integrability conditions are given by Yano (1965) and
others .
(1.12) CURVATURE TENSOR
Let D be the Riemannian connection on a Riemannian
manifold (Mn, g) . The Riemannian Christoffel curvature tensor of
second kind is defined as :
(1.12.1) ¢ R (X ,Y) Z = (D X D Y – D Y D X – D [X ,Y]) (Z) .
Let ¢R be the curvature tensor of type (0 , 4) given as :
(1.12.2) ¢R (X ,Y, Z , U) = g (R (X ,Y, Z) U ,
where ¢R is called the Riemannian Christoffel curvature tensor
of the first kind . It satisfies the following properties :
(i) Skew symmetric in first two slot
¢R (X ,Y, Z ,U) = -¢R (Y, X , Z ,U) .
(ii) Skew symmetric in last two slot
¢R (X ,Y, Z ,U) = -¢R (X ,Y,U, Z) .
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(iii) Symmetric in two pairs of slots
¢R (X ,Y, Z , U) = ¢R (Z , U , X ,Y) .
(iv) Bianchi’s first identities
¢R (X ,Y, Z ,U) = ¢R (Y, Z , X , U) + ¢R (Z , X ,Y,U) = 0
and (v) Bianchi’s second identities
(D X ¢R) (Y , Z , U , V) + (D Y ¢R) (Z , X , U ,V) +
(D Z ¢R) (X , Y, U ,V) = 0 .
(1.13) SPECIFIC CURVATURE TENSOR IN A RIEMANNIAN
MANIFOLD
(i) The Weyl - conformal curvature tensor ¢C
The Weyl - conformal curvature tensor ¢C of the type (0,4),
for N>3 is defined as follows :
(1.13.1) ¢C(X ,Y, Z,W) = ¢R(X ,Y, Z ,W) -2)-(n
1 [S(Y, Z) g(X ,W )
- S (X , Z) g (Y,W) + S(X ,W) g (Y, Z) –S(Y,W) g (X ,Z)]
+ 2)-1)(n-(n
r [g (Y, Z) g (X ,W) – g (X , Z) g (Y,W)] .
(ii) The Conharmonic curvature tensor ¢H .
The Conharmonic curvature tensor ¢H of the type (0,4) N>3 is
defined as :
(1.13.2)¢H (X ,Y, Z,W) = ¢R(X ,Y, Z ,W) –2)-(n
1 [S(Y, Z) g (X ,W)
- S (X , g (Y,W) + S (X ,W) g (Y, Z) – S (Y,W) g (X , Z)] .
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(ii) The projective curvature tensor ¢P .
The Projective curvature tensor ¢P of the type (0,4), for n>3 is
defined as follows :
(1.13.3) ¢P(X ,Y, Z,W) = ¢R(X ,Y, Z ,W) –1)-(n
1 [S(Y, Z) g (X ,W)
– S(X , Z) g (Y,W) .
(iv) The Concircular curvature tensor ¢V .
The Concircular curvature tensor ¢V of the type (0,4) , for
n>3 is defined as follows :
(1.13.4) ¢V(X ,Y, Z,W) =¢R (X ,Y, Z ,W) –1)-n(n
r [g(Y, Z) g(X ,W)
– g (X , Z) g (Y,W)].
The Concircular curvature tensor satisfies the following
properties :
(1.13.5a) ¢V(X ,Y, Z ,W) = –¢V(Y, X , Z ,W ) ,
(1.13.5b) ¢V (X ,Y, Z ,W) = –¢V(X ,Y,W, Z) ,
(1.13.5c) ¢V (X ,Y, Z ,W) = ¢V(Z ,W, X ,Y)
and
(1.13.5d) ¢V(X ,Y, Z ,W) +¢V(Y,Z , X ,W) +¢V(Z , X ,Y,W) = 0 .
(1.14) ALMOST COMPLEX MANIFOLDS
If Mn be n - dimensional C¥ differentiable manifold with a
non - zero tensor field f of type (1 , 1) satisfying :
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(1.14.1) F2 + I n = 0 ,
where I n is the identity matrix of order n , then Mn is called an
almost complex manifold and {F} is called an almost complex
structure of rank n . On an almost complex manifold Mn, a bilinear
function A is said to be
(1.14.2) Pure :
if A (X ,Y) + A (FX , FY) = 0
and
(1.14.3) Hybrid:
if A (X ,Y) = A (FX , FY) .
Let g be a positive definite Riemannian metric induced on an
almost complex manifold Mn satisfying :
(1.14.4) g (FX , FY) = g (X ,Y)
then the manifold Mn is called an almost Hermite manifold and the
structure {F , g} is called almost Hermite structure to Mn .
Let h be a fundamental 2- form defined by :
h (X ,Y) = g (F X ,Y )
then for an almost Hermite manifold , we have :
(1.14.5a ) h (X ,Y) = – h (Y, X) ,
(1.14.5b ) h (FX , FY) = h (Y, X)
and
(1.14.5c ) h (FX , FY) + h (X , FY) = 0
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The Nijenhuis tensor on an almost Hermite manifold is
defined as :
(1.14.6 ) N (X ,Y) = [FX , FY] – [X ,Y] – F[FX ,Y] – F [X , FY]
An almost complex manifold with the vanishing Nijenhuis
tensor is called a complex manifold .
An almost Hermite manifold Mn is reduces to the following
forms :
Kähler manifold if
(1.14.7 ) (D X F) (Y, Z) = 0.
Also , a Kaehler manifold can be reduced to Nearly Kähler
manifold or Tachibana manifold if
(1.14.8 ) (D X F) (Y, Z) = (D Y F) (Z , X) .
Almost Kaehler manifold if F is closed i.e.
(1.14.9 ) dF = 0 Þ
(D X F) (Y , Z) + (F Y F) (Z , X) + (D Z F) (X ,Y) = 0 .
Quasi Kaehler manifold if
(1.14.10 a ) (D X F) (Y, Z) + (D FX F) (FY, Z) = 0
and
(1.14.10b ) (D FX F) (Y, Z) = (D X F) (FY, Z) .
Hermite manifold if
(1.14.11) N (X ,Y, Z) = 0 Û
(D X F) (Z , X) = (D FX F) (FY, Z) – (D FY F) (FZ , X) .
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Nearly Hermite manifold if
(1.14.12) (D X F) (Y , Z) + (F Y F) (Z , X) = (D FX F) (FY , Z)
– (D FY F) (FZ , X) .
(1.15) ALMOST CONTACT METRIC MANIFOLDS
An odd dimensional differentiable manifold Mn with a real
vector valued linear function F, a 1-form η and a vector field x
called an almost contact manifold if it satisfies the following
conditions[39] :
(1.15.1a ) F2 X = – X + h (X) x ,
(1.15.1b ) h (x) = 1 ,
(1.15.1c ) h (FX) = 0
and
(1.15.1d ) rank (F) = n -1
for arbitrary vectors X ,Y , Z .[56]
The structure (F , x , h , g) is called an almost contact
structure on Mn . An almost contact manifold Mn with a metric
tensor g satisfying :
(1.15.2 ) g (FX , FY) = g (X ,Y) - h (X) h (Y)
and
(1.15.3 ) g (X , x) = h (X)
is called an almost contact metric manifold or an almost Grayan
manifold . The structure (F , x , h , g ) is called an almost contact
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metric structure [55]. The fundamental 2- form h of an almost
contact metric manifold is defiend by:
(1.15.4 ) h (X ,Y) = g (FX ,Y) .
Then ,we have :
(1.15.5 ) h (X ,Y) = h (FX , FY) and h (X ,Y) = - h (Y, X) .
If an almost contact metric manifold the fundamental 2 - form
h is such that
(1.15.6) 2 h (X ,Y) = (D X h) (Y) – (D Y h) (X)
then Mn is called Almost Sasakian manifold or a Contact
Riemannian manifold. An almost Sasakian manifold Mn in which
1-form h is a Killing vector [41], [48], i.e.
(1.15.7 ) (D X h) (Y) + (D Y h) (X) = 0
is called a K- Contact Riemannian manifold .
A manifold Mn is called a Sasakian manifold [39] if the
following relation holds true :
(1.15.8 ) (D X F) (Y) = h (Y) (X) – g (X ,Y) x
and
(1.15.9 ) h (X ,Y) = (D X h) (Y) and D X x = FX .
An almost contact metric manifold is can be reduced to
various structures as mentioned below :
(i) Co - symplectic manifold :
(1.15.10) if (D X F) (Y) = 0 and D X x = 0
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(ii) Nearly Co-symplectic manifold [6] :
(1.15.11) if (D X F) (Y) = (D Y F) (X).
(iii) Nearly Sasakian manifold [8]:
(1.15.12 ) if (D X F) (Y, Z) - h (Y) g (X , Z ) + 2h (Z) g (X ,Y)
= (D X F)(Z , X) + h (X) g (Y, Z).
(iv) Kenmotsu manifold [27] :
(1.15.13 ) if (D X F) (Y) = g (FX ,Y) x - h (Y) FX
and
(1.15.14 ) D X x = X - h (X) x.
(v) Nearly Kenmotsu manifold :
(1.15.15 ) If (D X F) (Y) + (D Y F) (X) = - h (Y) FX - h (X) FY.
(vi) Trans - Sasakian manifold [58] :
(1.15.16 ) if (D X F) (Y) = a{ g (X ,Y) x - h (Y) X}
+ b{g (Y, FX) x - h (Y) F X },
and
(1.15.17 ) D X x = - a F X + b{ X - h (X) x} ,
where a and b are non - zero constants .
(vii) Nearly Trans - Sasakian manifold :
(1.15.18 )if (D X F)(Y) + (D Y F)(X) = a{2g (X ,Y) x - h (Y) X
– h (X) Y} - b{h (Y) FX + F (X) FY}.
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Some of the new structure derived from the almost contact
metric manifolds [40] are as under given :
(i) Generalized nearly Co-sympletic manifold :
(1.15.19) if (D X F) (Y,Z) - h (Y)(D X h) (FZ)+h(Z)[(D X h) (FY)
+(D Y h) (FX)] = (D X F)(Z ,Y) + h (X) (D Y h) (FZ)
(ii) Generalized Quasi-Sasakian manifold ;
(1.15.20 ) if (D X F) (Y, Z) + (D Y F) (Z , X) + (D Z F) (X ,Y)
= h (X){(D Z h)(FY) – (D Y h) (FZ)} + h (Y){(D X h)(FZ)
– (D Z h) (FX)}+ h (Z){(D Y h) (FX) – (D X h) (FY)} .
(iii) Generalized Quasi-Sasakian manifold of first kind :
(1.15.21 ) if (D X F)(Y, Z) +(D Y F) (Z , X) + (D Z F) (X ,Y)
= h (X) (D Z h) (FY) + h (Y) (D X h) (FZ) + h (Z) (D Y h) (FX) .
(iv) Generalized normal manifold :
(1.15.22 ) if (D FX F) (FY, Z) = (D X F) (Y, Z) - h (Y)(D X h)(FZ)
+h (Z){(D X h)(FY) + (D FX h) (Y)}.
(v) Generalized almost contact normal matric manifold :
(1.15.23) if (D X F) (FY, Z) - h (Z){(D X h) (FY) + (D FX h) (Y)}
= (D X F) (Y,Z) - h (Y) (D X h) (FZ)}.
(vi) Generalized almost contact pseudo - normal metric manifold :
(1.15.24 ) if (D X F) (FY, Z) + (D X F) (Y, Z) - h(Z){(D FX h)(Y)
– (D X h) (FY)} - h (Y) (D X h) (FZ) = 0 .
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(vii) Generalized almost contact nearly normal metric
manifold :
(1.15.25 ) if (D FX F) (FY, Z) – (D X F) (Y, Z) + h (Y) (D X h) (Z)
- h (D FX h) (Y) + (D X h) (FY) + (D Y h) (FX) = (D FY F) (FZ , X)
– (D Y F) (Z , X) - h (X){(D FY h)(Z) + (D Y h) (FZ)}.
(viii) Generalized almost contact Pseudo - normal metric
manifold of the first class :
(1.15.26) if (D FX F) (FY, Z) + (D X F) (Y, Z) - h (Z) (D FX h) (Y)
- h (Y) (D Z h) (FX) = 0
(1.16) ALMOST PARA - CONTACT METRIC MANIFOLDS
Let us defined on an n - dimensional differentiable manifold
Mn consider a tensor field F of type (1 ,1) , 1- form h and a vector
field x satisfying the following condition :
(1.16.1a ) F2X = X + h (X) x ,
(1.16.1b ) h (x) = 1,
(1.16.1c ) F (x) = 0
and
(1.16.1d) h (FX) = 0 ,
then Mn is called an almost Para-Contact manifold [57] .
Let g be the Riemannian metric satisfying:
(1.16.2a ) g (FX ,FY) = g (X ,Y) - h (X) h (Y)
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25
and
(1.16.2b ) g (X , x) = h (X)
The structure (F , x , h , g) satisfying (1.16.1) , (1.16.2) is
called an almost para-contact Riemannian structure . The manifold
with such as structure is called an almost para - contact
Riemannian manifold .
If we define h (X ,Y) = g (FX ,Y) then we have :
(1.16.3 ) h (X ,Y) = h (Y, X) and h (FX , FY) = h (X ,Y) .
If in (Mn, g) the relations
(1.16.4a ) (D X h) (Y) – (D Y h) (X) = 0 ,
(1.16.4b ) dh (X ,Y) = 0 , i.e. h is closed ,
(1.16.4c ) (D X F) (Y) = - g (X ,Y) x - h (Y) X + 2h (X) h (Y) x ,
(1.16.4d ) (D X h) (Y) + (D Y h) (X) = 2 h (X ,Y)
and
(1.16.4e ) D X x = FX
hold , then (Mn, g) is called para - Sasakian manifold or briefly P
- Sasakian manifold . Further , if in (Mn, g) the following relation
holds .
(1.16.5) (D X h) (Y) = - g (X ,Y) + h (X) h (Y)
along with (1.16.1) , (1.16.2a) and (1.16.5) , such a manifold is
termed as Special para -Sasakian manifold or briefly SP -
Sasakian manifold .
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(1.17) LORENTZIAN PARA - CONTACT METRIC
MANIFOLDS
Let us consider an n - dimensional differentiable manifold Mn
endowed with a (1 , 1) tensor field F , a contraveriant vector field x
, a covariant vector h and a Lorentzian metric g satisfying
(1.17.1,a) F2 X = X + h (X) x ,
(1.17.1,b) h (x) = -1 ,
(1.17.1,c) g (FX , FY) = g (X ,Y) + h (X) h (Y)
and
(1.17.1,d) g (X , x) = h (X)
for arbitrary vector field X and Y. Such a manifold Mn is called a
Lorentzian para -contact manifold and the structure (F, x , h , g) is
called Lorentzian para - contact structure [34] . An LP-contact
manifold Mn is called a Lorentzian para - Sasakian manifold or
briefly LP - Sasakian manifold if it satisfies :
(1.17.2) (D X F) (Y) = g (X ,Y) x + h (Y) X + 2h (X) h (Y) x
and
(1.17.3) D X x = FX ,
where D denotes the operator of covariant differentiation with
respect to the Lorentzian metric g . The Lorentzian metric g makes
a time like unit vector field , i.e.g (x , x) = –1. It can easily seen
that in an LP- Sasakian manifold , the following relations hold true:
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27
(1.17.4,a) F (x) = 0 ,
(1.17.4,b) h(FX) = 0
and
(1.17.4,c) rank (F) = n -1
Let h (X ,Y) def g (FX ,Y)
Then the tensor field h is symmetric in nature i.e.
h (X ,Y) = h (Y , X) .
Since the 1-form h is closed in an LP - Sasakian manifold ,
there for we have :
(1.17.5) h (X ,Y) = (D X h) (Y) and h (X , x) = 0
for any vector fields X and Y
In an LP - Sasakian manifold the following relations satisfied
[33] :
(1.17.6,a) g (R (X ,Y) Z , x )= h (R (X ,Y) Z) – g (Y, Z) h (X)
– g (X , Z) h (Y) ,
(1.17.6,b) R (x , X) Y = g (X ,Y) x - h (X) x ,
(1.17.6,c) R (x , X) x = X + h (X) x ,
(1.17.6,d) R (X ,Y) x = h (Y) X - h (X) Y ,
(1.17.6,e) S (X , x ) = (n-1) h (X)
and
(1.17.6,f) S (FX , FY) = S (X ,Y) + (n-1) h (X) h (Y)
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for any vector field X ,Y, Z , where R (X ,Y) Z is the Riemannian
curvature tensor and S is the Ricci tensor .
An LP-Sasakian manifold Mn is said to be Lorentzian Special
Para-Sasakian (briefly LSP-Sasakian) manifold , if it satisfies :
(1.17.7) h (X ,Y) = Î {g (X ,Y) + h (X) h (Y)}, Î2 = 1
An LP - contact manifold is called an LP – Cosympletic manifold
if :
(1.17.8) D X F = 0 Û (D X F) (Y, Z) = 0
On this manifold , we have :
(1.17.9) (D X h) (Y) = 0 and D X x = 0 ,
An LP - contact manifold is called an LP - nearly co –
symplctic manifold , if [50] :
(1.17.10) (D X F) (Y) + (D Y F) (X) = 0 .
An LP-Sasakian manifold is called an h-Einstein manifold if
its Ricci tensor S is of the form :
(1.17.11) S (X ,Y) = a g (X ,Y) + bh (X) h (Y)
for any vector field X ,Y where a , b are scalar function on Mn
[74].
(1.18) HSU - STRUCTURE
An n - dimensional differentiable manifold Mn of class C¥ is
said to be endowed with HSU –structure if there exists a tensor
field f (¹ 0) of type (1 , 1) satisfying [12],[25] :
(1.18.1) f2 = ar I ,
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where ‘a’ is a non zero complex number, ‘r’ is a positive integer
and I denotes the unit tensor field .
(1.19) ALMOST CONTACT HYPERBOLIC METRIC
MANIFOLD
If in a differentiable manifold M , there exists a vector valued
linear function F, a 1 - form u and a vector field U satisfying :
(1.19.1) F2 = I + u Ä U , F U = 0
M is called almost contact hyperbolic manifold if :
(1.19.2) u o F = 0 , u (U) = -1
The triad {F, U , u } is called almost contact hyperbolic structure.
An almost contact hyperbolic manifold in which metric
tensor g satisfies
(1.19.3) g ( FX , FY) = - g (X ,Y) = u (X) u (Y) ,
is called almost contact hyperbolic metric manifold and the
structure {F,U , u , g} is called almost contact hyperbolic metric
structure [39] .
(1.20) SUBMANIFOLDS
Let M and M be two C¥ manifolds of dimension n and m (m
> n) respectively. A map i : M ® M is called an immersion if its
differential map i* . is injective for every x Î M . The image i(M) is
called imbedding of M into M . Manifold M is said to be a
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submanifold of M if it is a subset of M and the map i : M ® M is
both immersion and injective. If M is open , then the submanifold
is called an open submanifold .
Let M be a submanifold of a Riemannian manifold M with a
Riemannian metric g , then Gauss and Weingarten formulae are
given respectively as :
(1.20.1) X XY Y h (X ,Y)Ñ = Ñ +
and
(1.20.2) N, XA- N XNX^Ñ+=Ñ
for all X ,Y Î TM and N Î T^M , where Ñ , Ñ, Ñ^ are the
Riemannian connection , induced Riemannian connection and
induced normal connections in M , M and the normal bundle of M
respectively .
If h is the second fundamental form related to A as:
(1.20.3) g (h (X ,Y) , N) = g (A N X ,Y) .
If F is a (1 ,1) tensor field on M for X , Y Î TM and N Î
T^M , we put
(1.20.4) FX = PX + QX , PX Î TM , QX Î T^M
and
(1.20.5) FN = BN + CN , BN Î TM , CN Î T^M .
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31
The submanfold M is said to be totally geodesic if h = 0 and
totally umbilical if
(1.20.6) h (X ,Y) = g (X ,Y) H
(1.21) CR - SUBMANIFOLDS
Let M be an almost Hermitian manifold with almost complex
structure tensor F of type (1 , 1) . We consider a submanifold M of
M and the tangent space denote by TX M and the normal space
TX^M of M at x respectively. If TX M is invariant under the action
of F for each x Î M , that is if FTX M Ì TX M for each xÎM , then
M is called an invariant (or holomorphic) submanifold of M . On
the other hand, if the transformation of TXM by F is contained in
the normal space TX^M for each xÎM, that is FTX M Ì TX
^M for
each xÎM , then M is called an anti - invariant (or totally real)
submanifold of M .
Let M be an almost Hermitian manifold with almost complex
structure F. A submanifold M of M is called a CR - submanifold of
M if there exists a differentiable distribution D: x ® DX on M
satisfying the following conditions [4]:
(i) D is invariant , that is FDX Ì DX for each xÎM ,
and
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32
(ii) The complementary orthogonal distribution D^:x ® DX^
Ì TXM of D is anti-invariant, that is FDX^ Ì TX
^M for
each xÎM.
(1.22) SEMI – INVARIANT SUBMANIFOLDS
A semi - invariant submanifold is nothing but the extension
of the concept of the CR - submanifold of Kaehler manifold to
submanifold of almost contact metric manifolds .
Let M be submanifold of an almost contact metric manifold
M with almost contact metric structure (F , U , u , g). Then M is
called a semi - invariant submanifold of M if there exist two
differentiable distributions D and D^ on M satisfying.
(i) TM = D Å D^Å {U}, where D , D^ and {U} are mutually
orthogonal to each other.
(ii) The distribution D is invariant by F, that is F(DX) Ì DX for
each xÎM
and
(iii) The distribution D^ is anti-invariant by F, that is FDX^ Ì
TX^M for each x ÎM.
(1.23) HYPERSURFACES
A submanifold M of M is said to be a hypersurface of M if
dimension of M is one greater than dimension of M . In case of a
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33
hypersurface there is only one normal vector field to M . Let M be
an almost contact manifold and M be an orientabel hpersurface of
M . If i* is the differential of immersion i of M into M . Then for a
unit normal vector N and X , Y , Z tangent to M we have :
(1.23.1) (a) FBX = Bf X + u (X) N , (b) FN = - BU
(1.23.2) and g (BX , BY) ob = h(X,Y),
where f is a tensor field of type (1,1) , u is a 1-form , U is a vector
field and h is a induced metric tensor on M.
If u is identically zero , then M is said to be an invariant
hypersurface . In the other words the tangent space of M is
invariant by F. If u ¹ 0, then M is called a non-invariant
hypersurface of M .
Let E be the induced metric connection on the hypersurface
M. Then we have :
(1.23.3) (a) D BX BY = BE XY + ¢H (X ,Y) N ,
and (b) D BX N = - BHX ,
where ¢H is the second fundamental tensor on M and H is the
associate tensor given by
(1.23.4) ¢H (X ,Y) = h (HX ,Y).
(1.24) THE TANGENT BUNDLE
Let M be an n - dimensional differentiable manifold of class
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34
C¥ and Tp(M) be the tangent space of M at pÎM , then expression
(1.24.1) (M) T T(M) pM pÎ
= U
is called the tangent bundle of the manifold M. If (M)T p~ pÎ , then
the correspondence p p~® determines the bundle projection
p:T(M) ® M . Thus ( ) p p~π = and the set p-1(p) is called the fibre
over pÎM and M is called the base space .
Suppose the base space is covered by a system of coordinate
neighbourhoods (U, xh) where h takes the values 1 to n , then (xh)
is the system of local coordinates defiend in the neighbourhood U.
Let (U¢, xh¢) be another coordinate neighbourhood in M
containing the point p . Then p-1(U¢) contains p~ and the induced
coordinates of p with respect to p-1(U¢) will be given by (xh¢, yh¢)
where
(i) xh¢ = xh (x)
(1.24.2) and
(ii) h
h
hh y
x'x
'y¶¶
=
Such that xh¢(x) is differentiable functions of class C¥ of variables
x1,x2, ….xn and the derivatives being evaluated at p.[73]
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(1.25) THE COTANGENT BUNDLE
If M be an n-dimensional C¥ manifold of (M)T*p , then
(1.25.1) (M)T (M)T *m
M p
*
Î= U
is called the cotangent bundle of the manifold M . For any point
(M)T p~ *mÎ the correspondence p p~® determines the bundle projection
p:T*(M)®M.
Let M be covered by a system of coordinates neighbourhoods
{U , xh} and xh is the system of local coordinates defined in U. If
{U¢, xh¢} be other coordinates neighbourhood in M containing the
point p, then p-1(U¢) contains p~ and the induced coordinates in p-
1(U¢) are (xh¢, pi¢) given as :
(i) xh¢ = xh¢(x)
(1.25.2) and
(ii) ii'
i
i' pxx
p¶¶
=
where xh¢(x) are differentiable coordinates of class C¥ and n
variables x1,x2, ….xn and the derivatives evaluated at p .
(1.26) THE COMPLETE AND HORIZONTAL LIFTS IN THE COTANGENT BUNDLE
(i) Complete lift
Let (M)J rs be the set of tensor fields of type (r , s) of class C¥
in M and r *sJ (T (M)) be the corresponding set of tensor fields in
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36
T*(M) . Suppose now that F (M)εJ11 and F has components h
iF at
point A in the coordinate neighbourhood U. Then at (A , p) in
p-1(U) , we can define 1- form s as :
(1.26.1) sI = pa aiF ; i
s = 0 ,
Thus
(1.26.2) s = pa aiF dxi .
The exterior derivative ds of s is a 2-form given by :
(1.26.3) e
a b a bbbC
Fd pa dx dx F dpa dx
xsss
= L + L .
If we write
(1.26.4) ds = B1 BC dx
2t Ù dxc ,
we have :
a
iiji ji ji pa ( - ); F
xj x
bjFFt t
¶¶= =
¶ ¶
where L is Skew-symmetric .
(1.26.5) ijji ji
- F ot t= = .
The tensor field of type (1,1) in T*(M) is denoted by FC
where components ABF
~ in p-1(U) are given by
(1.26.6) A C A
B BCF t x=% ,
where xCA are components of (2,0) type tensor x in p-1(U) .
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37
Thus ,
o F~
;F F~ h
ih
ih
i ==
and
(1.26.7) ih
hih
ai
i
ahh
i F F~ );
x
F -
x
F( pa F
~ =¶¶
¶¶
=
FC is called complete lift of the tensor field F .
(ii) The Horizontal lift
Let Ñ be symmetric affine connection in M and U, U¢ be the
coordinate neighbourhoods containing the point A of M. Suppose
that F Î 11J and F has components h
iF at A in a neighbourhood U of
M . Let Ñ have components hjiG and h'
jiG relative to U, U¢
respectively at A and p have components pi and pi¢ relative to U
and U¢ respectively. Then jiG and 'jiG are defiend as
(1.26.8) a ' a'
ji a ji ji a' ji p ; p G = G G = G .
The components ABF
~ relative to p-1(U) of the tensor field of a type
(1 , 1) at the point (A , p) in T*(M) are given by :
o F~ ;F F
~ hi
hi
hi ==
(1.26.9) ih
hi
aiha
ahia
hi F F
~ ;F F - F
~=G+G=
Also , the components A'BF
~ relative to p-1(U¢) are given by :
o 'F~ ;F' 'F
~ hi
hi
hi ==
(1.26.10) ih
hi
ai
'ha
ah
'ia
hi F' 'F
~ ;F' F' - 'F
~=G+G=
This tensor field is denoted by FH and is called the Horizontal
lift of F [73] .
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