CHAPTER 1 PRELIMINARIES

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1 CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable manifolds Let M be a topological space. We assume that M satisfies the Hausdorff separation axiom which states that any two different points in M can be separated by disjoint open sets. An open chart on M is a pair U, where U is an open subset of M and is a homeomorphism of U onto an open subset of n,where n is an n-dimensional Euclidean space. Definition (1.2.1) Let M be a Hausdorff space. A differentiable structure of dimension n is a collection of open charts U i, i i I on M where i U i is an open subset of n such that the following conditions are satisfied: (i) M i I U i, (ii) The mapping j 1 i is a differentiable mapping of i U i U j onto j U i U j for each pair i, j I, (iii) The collection U i, i i I is a maximal family of open charts which satisfy the conditions (i) and (ii). The Hausdorff topological space M with a differentiable structure is called a smooth (orc or differentiable) manifold of dimension n. 1

2 In order to define a complex manifold of (complex) dimension n, we replace n in the definition of differentiable manifold by n-dimensional complex number space C n. The condition (ii) is replaced by the condition that the n-coordinates of j 1 i p should be holomorphic function of the co-ordinates of p. Definition (1.2.2) Let M be an n-dimensional smooth manifold, p M.A tangent vector X p to M at p is a function X p : C p, whose value at f is denoted by X p f such that for f, g C p, and,, the following conditions are satisfied: (i) X p f g X p f X p g, (ii) X p fg X p f g p f p X p g. Definition (1.2.3) The set of all tangent vectors at p M and we call it the tangent space at p. is denoted by T p M Definition (1.2.4) Let f : M be a smooth map. Then for p M we define the differential df p of f at p as df p : T p M by df p X p X p f Definition (1.2.5) Let f : M N be a smooth map. Then for p M the differential df p of f at p is the map df p : T p M T f p N defined by df p X p g X p f g X p g f where g C f p, X p T p M Definition (1.2.6) Let f : M N be a smooth map. Then f is called (i) an immersion if df p : T p M T f p N is one-to-one map, p M. (ii) an imbedding if f is a one-to-one immersion. (iii) a diffeomorphism if f is a bijection and f 1 is smooth. (iv) a submersion if f is onto and df p : T p M T f p N is onto, p M. 2

3 Definition (1.2.7) Let M be a smooth manifold of dimension n. Avector field X on M is a rule which associates an element X p T p M with every p M. In term of local co-ordinates x i, a vector field X maybeexpressedas X X i where X i are functions defined in the co-ordinates x i neighborhood and are called components of X with respect to x i. A vector field X is differentiable if and only if its components are differentiable. The set of all C vector fields on a differentiable manifold M forms a real vector space under the natural addition and scalar multiplication. This vector space is denoted by M. If X, Y are C vector fields, then we define a C vector field called the Lie bracket of X and Y by X, Y XY YX. If X X i and Y Y j, then X, Y f X i Yj Y i Xj f x i x j x i x i x j. This shows that X, Y is a vector field whose components relative to x i are X i Yj x i Y j Xj x j. Theorem (1.2.8) If X, Y, Z M and f, g C M, a, b, then (i) X, Y Y, X (ii) ax by, Z a X, Z b Y, Z (iii) fx, gy fg X, Y g Yf X f Xg Y (iv) X, Y fg f X, Y g g X, Y f (v) X, Y, Z Y, Z, X Z, X, Y 0 (Jacobi identity) (vi) X, X 0 Let T p M be the daul space of the tangent space T p M of M at p. An element of T p M is called a covector at p. An assignment of a covector at each point p is called a 1-form. Every 1-form can be uniquely written as 3

4 i f i dx i. where f i respect to x i. are functions called the components of with 1.3 Riemannian manifolds Definition (1.3.1) A linear connection on a differentiable manifold M is a mapping : M M M, which is denoted by X, Y X Y and satisfies the following properties: (i) fx gy Z f X Z g Y Z, (ii) X Y Z X Y X Z, (iii) X fy f X Y X f Y where X, Y, Z M and f, g C M. The operator X is called the covariant differentiation with respect to X. The covariant differentiation of a function f with respect to X is defined by X f Xf. The torsion tensor T of a linear connection is a tensor field T of type (1,2) defined by T X, Y X Y Y X X, Y. Definition (1.3.2) A Riemannian metric (or Riemannian structure) ona differentiable manifold M is a correspondence which associates to each point p of M an inner product g which is symmetric, bilinear, positive definite form on the tangent space T p M. A differentiable manifold with a given Riemannian metric is called a Riemannian manifold. The curvature tensor R of a linear connection on a manifold M is a tensor field of type (1,3) i.e., R : M M M M defined by R X, Y Z X Y Z Y X Z X,Y Z, for any X, Y, Z M. 4

5 and Riemannian curvature tensor is given by R X, Y, Z, W g R X, Y Z, W for any X, Y, Z, W M. Proposition (1.3.3) The curvature tensor R satisfies the following: (i) R X, Y, Z, W R Y, X, Z, W, (ii) R X, Y, Z, W R X, Y, W, Z, (iii) R X, Y, Z, W R Z, W, X, Y, (iv) R X, Y, Z, W R Y, Z, X, W R Z, X, Y, W 0, (v) R X, Y, Z, Z 0 for any vector field X, Y, Z, W M. Theorem (1.3.4) ( The fundamental theorem of Riemannian geometry) Given a Riemannian manifold M, there exists a unique connection on M satisfying the conditions: (i) is symmetric, T 0 i.e, X Y Y X X, Y for all X, Y M. (ii) is compatible with the Riemannian metric. i.e, Xg Y, Z g X Y, Z g Y, X Z for all X, Y, Z M. The connection given by the theorem is called Levi-Civita (or Riemannian) connection on M. Definition (1.3.5) For each plane in the tangent space T p M the sectional curvature K for is defined by K R U, V, V, U g U, U g V, V g U, V 2 where U, V. For orthonormal vector fields U, V on we have K R U, V, V, U. 5

6 1.4 Almost Hermitian manifolds In 1947, A.Weil observed the presence of almost complex structure J on complex manifolds. This structure was introduced on an even dimensional real manifold by Ehresmann thus convincing the geometers that this additional structure on Riemannian manifolds helps to know the geometry of manifolds in great depth. This led to the new branch in geometry called geometry of manifolds with structures. Let M be a real differentiable manifold. A tensor field J on M is called an almost complex structure on M if at every point p of M, J is an endomorphism of the tangent space T p M such that J 2 I. AmanifoldM with a fixed almost complex structure J is called an almost complex manifold. Every almost complex manifold is of even dimensional. A Hermitian metric on an almost complex manifold M is a Riemannian metric g such that g JX, JY g X, Y for X, Y M. An almost complex manifold (resp. a complex manifold) with Hermitian metric is called an almost Hermitian manifold (resp. a Hermitian manifold). Let M be an almost Hermitian manifold with almost complex structure J and Hermitian metric g. The fundamental 2-form of M is defined by X, Y g X, JY for all vector field X, Y on M. Then we have JX, JY X, Y. The torsion tensor field N of type (1,2) of an almost complex structure J called Nijenhuis tensor is defined by N X, Y JX, JY J X, JY J JX, Y X, Y for any vector field X, Y on M. 6

7 Definition (1.4.1) An almost Hermitian manifolds M is said to be: (i) Kaehler if X J Y 0, (ii) Nearly Kaehler if X J Y Y J X 0, (iii) Quasi Kaehler if X J Y JX J JY 0, (iv) Hermitian if N X, Y 0, for any vector field X, Y on M. The holomorphic bisectional curvature of an almost Hermitian manifold M is defined for any pair of unit vectors E and F on M by B E, F g R E, JE JF, F. Then the holomorphic sectional curvature of M is given by H E B E, E. 1.5 Almost contact metric manifolds Definition (1.5.1) Let M be an 2n 1 -dimensional real differentiable manifold and, and be a tensor field of type 1, 1, a vector field and a 1-form respectively on M.If, and satisfy the conditions 2 I, 1 (1.5.1) where I denotes the identity transformation. Then M is called an almost contact manifold and,, the almost contact structure on M.From (1.5.1), it follows that 0, 0. 7

8 Moreover, every almost contact manifold M metric tensor field g such that admits a Riemannian g X, Y g X, Y X Y, g X, X where X and Y are vector fields on M. Then M is called an almost contact metric manifold and,,, g the almost contact metric structure. The torsion tensor field N of of type (1,2) called Nijenhuis tensor is defined by N X, Y X, Y X, Y X, Y X, Y, for any vector field X, Y on M. Definition (1.5.2) An almost contact metric manifold M,,, g is said to be: (i) Normal if N X, Y d X, Y 0, (ii) Sasakian if X Y g X, Y Y X, X X, (iii) cosymplectic if X Y 0, (iv) quasi-k-cosymplectic if X Y X Y Y X, (v) Kenmotsu if X Y g X, Y Y X, X X X, (vi) trans-sasakian if X Y g X, Y Y X g X, Y Y X, X X X X, for any vector field X and Y. A trans-sasakian structure is of type, where and are non-zero constants. If 0 then M is a -Kenmotsu manifold and if 0 then M is a -Sasakian manifold. If 1, 0 (resp. 0, 1), then M is a Sasakian manifold (resp. Kenmotsu manifold). Moreover, if 0, 0, then M is called cosymplectic manifold. 8

9 Let M be an almost contact metric manifold with structure,,, g. We denote by B the -holomorphic bisectional curvature, defined for any pair of non zero vectors E and F on M orthogonal to by the formula 19 B E, F E 2 F 2 R E, E, F, F. The -holomorphic sectional curvature of M is given by H E B E, E for any nonzero vector E orthogonal to. 1.6 Theory of submanifolds Definition (1.6.1) Given two smooth manifolds M and M of dimension n, n K respectively, if there exist a smooth immersion : M M then we say that M is a submanifold of M.If is an imbedding then M is said to be an imbedded submanifold of M. Let M be a submanifold of a Riemannian manifold M. The Riemannian metric g of M induces a Riemannian metric on M which we denote by the same letter g. For each p M, the inner product on T p M splits T p M into the direct sum T p M T p M T p M where T p M is the orthogonal complement of T p M in T p M. The Riemannian connection of M induces Riemannian connection and in the tangent bundle TM and in the normal bundle T M respectively,andtheyarerelatedbythegauss and Weingarten formulas X Y X Y h X, Y and X N A N X X N (1.6.1) (1.6.2) where X, Y TM and N T M, h X, Y and A N X are second 9

10 fundamental form and second fundamental tensor related by g A N X, Y g h X, Y, N. Let the curvature tensor corresponding to and be denoted by R and R respectively. Then the Gauss equation is given by R X, Y, Z, W R X, Y, Z, W g h X, W, h Y, Z g h X, Z, h Y, W. (1.6.3) Where X, Y, Z, W TM.TheCodazzi equation is given by R X, Y Z X h Y, Z Y h X, Z. where X h Y, Z X h Y, Z h X Y, Z h Y, X Z for X, Y, Z TM Definition (1.6.2) An m-dimensional distribution on a manifold M is a mapping D defined on M which assigns to each point p M an m-dimensional linear subspace D p of T p M i.e., D : p M D p T p M. A vector field X on M belongs to the distribution D if we have X p D p for each p M and when this happens we write X D. Definition (1.6.3) A distribution D is said to be integrable (or involutive) if for all vector fields X, Y D we have X, Y D. Definition (1.6.4) Let be a connection on M. The distribution D is said to be parallel with respect to if we have X Y D for any X, Y D. Definition (1.6.5) The submanifold M is said to be totally geodesic in M if its second fundamental form vanishes identically, i,e., h 0. Let M be a n-dimensional almost Hermitian manifold with almost complex structure J and with Hermitian metric g. Let M be a real m-dimensional Riemannian manifold isometrically immersed in M.The differential geometry of M depends on the behaviour of the tangent bundle 10

11 of M relative to the action of the almost complex structure J. Thus M is called a holomorphic (or invariant) submanifold if T p M is invariant by J i.e., we have J T p M T p M for each p M. Also, we say that M is a totally real (or anti-invariant ) submanifold of M if we have J T p M T p M for each p M These two classes of submanifolds have been extensively investigated from different points. The fundamental result on the geometry of totally real submanifold can be found in Yano-Kon 37. In 6 A. Bejancu introduced the notion of a new class of submanifold as a natural generalization of invariant submanifold and anti-invariant submanifold called Cauchy-Riemann (CR)-submanifold. Definition (1.6.6) Let M be an almost Hermitian manifold and M be a Riemannian manifold isometrically immersed in M. Then M is called a CR-submanifold of M if there exists a distribution D : p D p T p M such that : (i) the distribution D is holomorphic (or invariant) i.e., JD p D p for each p M, (ii) the complementary orthogonal distribution D : p D p T p M of D is anti-invariant, i.e., JD p T p M for each p M. If dimd p 0, (resp. dimd p 0) for each p M, then the CR-submanifold is a holomorphic ( resp. totally real ) submanifold. A CR-submanifold is called proper CR-submanifold if it is neither holomorphic nor totally real. Let the orthogonal complement of JD T M be. Then we have T p M D p D p, T p M JD p p 11

12 Definition (1.6.7) A CR-submanifold M of an almost Hermitian manifold M is said to be D-totally geodesic (resp.d -totally geodesic )if h X, Y 0 resp. h Z, W 0 for all X, Y D Z, W D. M is said to be mixed totally geodesic if h X, Z 0 for any vector field X D and Z D. 1.7 Geometry of submersion The study of Riemannian submersion : M M of a Riemannian manifold M onto a Riemannian manifold M was initiated by B. O Neill 28. A submersion naturally gives rise to two distributions on M called the horizontal and vertical distributions respectively, of which the vertical distribution is always integrable giving rise to the fibers of the submersion which are closed submanifolds of M 17. Definition (1.7.1) Let M, g and M, g be two Riemannian manifolds. A smooth onto map : M M is said to be a Riemannian submersion if the following axioms are satisfied : (i) is a submersion, that is d p : T p M T b M is onto, p M, p b for all b M Because of submersion, for each b M, F b 1 b is a smooth submanifold of M, ofdimension dimm dimm, which is called fiber. Also p M with p b, T p M T P F b h P,whereh P T p M such that dimh P dimm and we call it horizontal subspace which is always orthogonal to the fibers, and T P F b p is a vertical subspace which is always tangent to the fibers. (ii) d perserves length of horizontal vectors. That is, g X p, Y p g d p X p, d p Y p, X p, Y p h P 12

13 Remark (1.7.2) For any vector field E we have E h E E is a unique representation where h E, E are horizontal and vertical components of E. The aim of the study of Riemannian submersion : M M is to relate the geometry of M to that of M. For this reason we choose a special vector field in computations with tensor equations which suits our purpose. Definition (1.7.3) A vector field X on M is said to be basic vector field if (i) X h i.e. X is horizontal. (ii) There is a vector field X on M such that d X X, that is X is -related to X. Remark (1.7.4) For vertical vector fields V and W, set V W V W, which is the induced connection on fibers as submanifold of M. Definition (1.7.5) The second fundamental form of all fibers 1 b gives rise to a tensor field T on M defined for arbitrary vector fields E and F by T E F T E, F h E F E hf where is the covariant derivative of M. T has the following properties: (i) At each point, T E is a skew-symmetric linear operator on the tangent space of M, and it reverses the horizontal and vertical subspaces. (ii) T is vertical, that is, T E T E. (iii) For vertical vector fields, T has the symmetry property that is T V W T W V Similarly, the tensor field A E F is defined by A E F A E, F he hf h he F. Again A has the following properties : (i ) At each point, A E is a skew-symmetric linear operator on the tangent space of M, and it reverses the horizontal and vertical subspaces. (ii ) A is horizontal, that is, A E A he. 13

14 (iii )For horizontal vector fields, A has the alternation property that is A X Y A Y X. The covariant derivative T and A will appear in the fundamental equations of submersion, so we need the expressions for it. Definition (1.7.6) For arbitrary vector fields V, W and F define (i) V A W F V A W F A V W F A W V F, (ii) V T W F V T W F T V W F T W V F. Remark (1.7.7) 19 (i) If X is a horizontal vector field and V is a vertical vector field then T X 0,A V 0. (ii) The restriction of A to h M h M measures the integrability of horizontal distribution. It is known that A 0 if and only if h is integrable. (iii) The restriction of T to v M v M acts as the second fundamental form of any fibre. In particular, the vanishing of T means that any fibre of is totally geodesic submanifold of M. Lemma (1.7.8) 28 If X and Y are basic vector fields on M, then (i) g X, Y g X, Y, (ii) h X, Y is the basic vector field corresponding to X, Y, (iii) h X Y is the basic vector field corresponding to X Y, where is a Riemannian connection. (iv) for any vertical vector field V, X, V is vertical. Lemma (1.7.9) 28 If X, Y are horizontal vector fields, then A X Y A Y X 1 X, Y. 2 (1.7.1) 14