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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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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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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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