Lie Derivatives and Almost Analytic Vector Fields in a Generalised Structure Manifold

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 2, Lie Derivatives and Almost Analytic Vector Fields in a Generalised Structure Manifold R. P. Singh 1 and S. D. Singh Dept. of Mathematics, Faculty of Science Banaras Hindu University, Varanasi , India Abstract The purpose of the present paper is to study the properties of Lie derivatives and almost analytic vector fields in a Generalised Structure Manifold. In this paper certain theorems are also have been proved which are of great geometrical importance. Mathematics Subject Classification: 53C15, 53C21, 57P05 Keywords: C -manifold, Generalised structure manifold, Tangent structure manifold, Almost Hermite manifold, Metric π-structure manifold, Hsustructure manifold, F -structure manifold, Lie derivative, Almost analytic vector field 1 Introduction We consider a differentiable manifold V n of differentiability class C and of dimension n. Let there exist in V n a tensor field F of the type (1, 1), s linearly independent vector fields U i, i =1,2,..., s and s linearly independent 1-forms u i such that for any arbitrary vector field X, we have X = b 2 X + cu i (X)U i (1.1) U i = p j i U j (1.2) where F (X) def = X and b 2, c are constants. Then the structure {F, u i, U i, p j i ; i,j =1,2,..., s} will be known as generalised structure and V n will be known as generalised structure manifold of order s where s<n. Agreement 1.1. All the equations which follow hold for arbitrary vector fields X, Y, Z,... etc. 1 rajabhaia@gmail.com, pratap ravindra@hotmail.com

2 82 R. P. Singh and S. D. Singh Now, replacing X by X in (1.1),we get X = b 2 X + cu i (X)U i (1.3) Operating F on both sides of (1.1) and using (1.2) we get From (1.3) and (1.4), we have X = b 2 X + cu i (X)p j i U j (1.4) u i (X) =p i ju j (X) (1.5) Further, operating F in (1.2) and using (1.1), (1.2) we get where (2) p j i = b2 δ j i + cuj (U i ) (1.6) (r) p i j =(r 1) p i k pk j On generalised structure manifold V n, let us introduce a metric tensor g such that 2-form F defined by F (X, Y ) def = g(x,y ) is skew-symmectric,then V n is called generalised metric structure manifold.([6]) We have on a generalised metric structure manifold, g(x,y )+g(x, Y )=0 Replacing Y by Y in above equation and using (1.1), we obtain where g(x,y )+b 2 g(x, Y )+cu i (X)u i (Y ) = 0 (1.7) u i (X) =g(u i,x) (1.8) Agreement 1.2. The generalised metric structure manifold always be denoted by V n. 1.1 Definitions This section consists of well known definitions required to go through the insuring sections.([1], [4]) 1. A differentiable manifold M n on which there a vector valued linear function F, a 2-form F defined by F (X, Y ) def = g(x,y ) such that F 2 = 0 and F (X, Y ) is skew-symmetric, then M n is called an almost tangent metric manifold.

3 Lie derivatives and almost analytic vector fields 83 F 2 = I n and F (X, Y ) is skew-symmetric, then M n is called an almost Hermite manifold. F 2 = λ 2 I n, where λ is a non-zero complex constant and F (X, Y )is skew-symmetric, then M n is called an metric π-structure manifold. F 2 = λ r I n and F (X, Y ) is skew-symmetric, then M n is called an Hsu-structure metric manifold. F 2 = I n and F (X, Y ) is symmetric, then M n is said to be an almost product Riemannian manifold. 2. Let us consider a C -manifold M n (n =2m + 1). Let there exist in M n a tensor field F of the type (1, 1), a 1-form u, a vector field U and a Riemannian metric g satisfying X = X + u(x)u U = 0 g(x,y )=g(x, Y ) u(x)u(x) (1.9a) (1.9b) (1.9c) where g(x, U) =u(x) and F (X) def = X Then M n is called an almost contact metric manifold or an almost Grayan manifold. 3. We consider a manifold M n of differentiability class C. Let there exist in M n, a tensor field F of the type (1, 1) and rank r (1 r n) satisfying F 3 + F = 0 (1.10) then {F } is called F -structure and M n satisfying (1.10) is called F - structure manifold. If we consider F (X, Y ) def = g(x,y ) where g is a Riemannian metric and F is skew-symmetric then, F -structure manifold M n is called a metric F -structure manifold. 4. A vector Field U i in a generalised structure manifold said to be a Killing vector if it satisfies ([2], [5]) (D X u i )(Y )+(D Y u i )(X) = 0 (1.11) and U i is said to be a Harmonic vector if it satisfies (D X u i )(Y ) (D Y u i )(X) = 0) (1.12a) and (D X U i ) = 0 (1.12b)

4 84 R. P. Singh and S. D. Singh Remark 1.1. It may be noted that V n gives an almost tangent metric manifold, an almost Hermite manifold, metric π-structure manifold, Hsu-structure manifold, F -structure manifold, an almost product Riemannian manifold, an almost Grayan manifold and { F, g, u 1, u 2, U 1, U 2 } structure manifold according as (b 2 =0,c =0); (b 2 = 1,c =0); (c =0); (b 2 = λ r,c =0); (b 2 = 1,p j i =0); (b2 =1,c =0); (b 2 = 1,c =1,p 1 1 =0:i, j =1); and (b 2 = 1,c=1,p j i + pi j =0:i, j =1, 2) respectively. 2 Lie Derivatives Let X be a C -vector field on an open set A. Lie derivative via. X is a type preserving mapping L X : T r s T r s such that ([3]) where u is a 1-form, and L X f = Xf, f is a C -function (2.1) L X a = 0, a R (2.2) L X Y = [X, Y ] (2.3) (L X u)(y ) = Xu(Y ) u([x, Y ]) (2.4) (L X P )(α 1,α 2,...,α r,x 1,X 2,...X s )=XP(α 1,α 2,...,α r,x 1,X 2,...X s ) P (L X α 1,α 2,...,α r,x 1,X 2,...X s ) P (α 1,α 2,...,L X α r,x 1,X 2,...X s ) P (α 1,α 2,...,α r,l X X 1,X 2,...X s ) P (α 1,α 2,...,α r,x 1,X 2,...L X X s ) (2.5) where P is a tensor of the type (r, s). With this definition, we at once have L X (P + Q) =L X P + L X Q ; P, Q T r s (2.6) L X (P R) =(L X P ) R + P (L X Q); P Ts r,r T u t (2.7) Theorem 2.1. In V n, with U i as a killing vector, we have (L X u i )(Y )+(L Y u i )(X) =u i (D X Y + D Y X) (2.8) Proof. From (2.4), we have (L X u i )(Y )=(D X u i )(Y )+u i (D Y X) (2.9a) Interchanging X and Y in above equation,we get (L Y u i )(X) =(D Y u i )(X)+u i (D X Y ) (2.9b) Adding (2.9a) and (2.9b) and using (1.11), we get (2.8).

5 Lie derivatives and almost analytic vector fields 85 Corollary In V n, we have (L X u i )(Y )+(L Y u i )(X) =u i ((D X F )(Y )+(D Y F )(X)) + p i j uj (D X Y + D Y X) (2.10) Proof. Replacing X by X and Y by Y in (2.8) and using (1.5), we get (2.10). Theorem 2.2. In V n, with U i as a harmonic vector, we have (L X u i )(Y ) (L Y u i )(X)+u i ([X, Y ]) = 0 (2.11) Proof. Subtracting (2.9b) from (2.9a) and using (1.12), we get (2.11) Now, let us consider Nijenhuis tensor N(X, Y ), which is a vector valued, bilinear function and given by N(X, Y ) def =[X,Y ]+[X, Y ] [X,Y ] [X, Y ] (2.12) For the symmetric connexion D, it becomes N(X, Y )=D X Y D Y X + D X Y D Y X D X Y + D Y X D X Y + D Y X (2.13) Using (1.1) in above, we get N(X, Y )=D X Y D Y X + b 2 (D X Y D Y X)+cu i (D X Y D Y X)U i Theorem 2.3. In V n, we have D X Y + D Y X D X Y + D Y X (2.14) (L X F )(Y )+(L X F )(Y )=N(X, Y )+c{u i (Y )[X, U i ]+(u i (D Y X)+(D X u i )(Y ))U i } (2.15) Proof. We have, (L X F )(Y )=D X Y D Y X D X Y + D Y X (2.16) Replacing X by X and Y by Y alternately in the above equation, we get (L X F )(Y )=D X Y D Y X D X Y + D Y X (2.17a) (L X F )(Y )=D X Y D Y X D X Y + D Y X (2.17b) respectively. (2.15). Adding (2.17a) and (2.17b) and using (1.1) & (2.14), we get

6 86 R. P. Singh and S. D. Singh Corollary In V n, we have (L X F )(Y ) (L X F )(Y )=N(X, Y ) (2.18) (L X F )(Y )+p i j (L XF )(U i )=N(X, U j )+( (2) p i j b2 δj i )[X, U i]+cu i ([U j,x])u i (2.19) Proof. Operating F on both sides of (2.16) and subtracting the resulting equation from (2.17a), we get (2.18) after using (2.13). Replacing Y by U j in (2.15) and using (1.6), we obtain (2.19). Theorem 2.4. In V n, with U i as a harmonic vector, we have p j i (L U j F )(Y )+(L Ui F )(Y )=N(X, U i ) (2.20) Proof. Replacing X by U i in (2.15) and using (1.2) & (1.12), we get (2.20). Now, corresponding to the Nijenhuis tensor of an almost complex manifold, we have three tensors μ, ν and σ given by def μ(x, Y ) = (D Y u i )(X) (D X u i )(Y )+(D Y u i )(X) (D X u i )(Y ) (2.21) def ν(x) = D Ui X (D X U i )+D X U i D Ui X (2.22) def σ(x) = (D X u j )(U i ) (D Ui u j )(X) (2.23) respectively. Theorem 2.5. In V n, we have μ(x, Y )=(L Y u i )(X)+(L Y u i )(X) (L X u i )(Y ) (L X u i )(Y ) +u i (D Y X D X Y +(D Y F )(X) (D X F )(Y )) p i j uj ([X, Y ]) (2.24) ν(x) (L Ui F )(X) = 0 (2.25) σ(x)+(l Ui u j )(X) = 0 (2.26) Proof. Replacing Y by Y in (2.9a), we get (L X u i )(Y )=(D X u i )(Y )+u i (D Y X) (2.27a) Replacing X by X in (2.9a) and using (1.5), we get (L X u i )(Y )=(D X u i )(Y )+u i ((D Y F )(X)) + p i j uj (D Y X) (2.27b) Interchanging X and Y in (2.27a) and (2.27b) separately, we have (L Y u i )(X) =(D Y u i )(X)+u i (D X Y ) (2.27c) (L Y u i )(X) =(D Y u i )(X)+u i ((D X F )(Y )) + p i ju j (D X Y ) (2.27d) respectively. Subtracting (2.27a) and (2.27b) from the sum of (2.27c) and (2.27d) and using (2.21), we get (2.24). putting U i for X and X for Y in (2.16) and using (2.22), we obtain (2.25). (2.26) can be obtained with the help of (2.9b), (1.6) and (2.23).

7 Lie derivatives and almost analytic vector fields 87 3 Almost Analytic Vector Fields A vector field V is said to be contravariant almost analytic in V n,ifit satisfies ([7], [8]) (L V F )(X) = 0 (3.1a) and (L V u i )(X) = 0 (3.1b) A vector field V is said to be strictly contravariant almost analytic in V n, if both V and V are contravariant almost analytic, i.e. if (3.1) and are satisfied. (L V F )(X) = 0 (3.2a) and (L V u i )(X) = 0 (3.2b) Theorem 3.1. The condition that a vector V be contravariant almost analytic in V n,is [V,X] =[V,X] and (D V u i )(X)+u i (D X V ) = 0 (3.3) Proof. In consequence of (2.3) and (2.4), we have (L V F )(X) =[V,X] [V,X]and (L V u i )(X) = (D V u i )(X)+u i (D X V ) (3.4a) By virtue of (3.1a) and (3.1b), we get (3.3). Remark 3.1. If V is contravariant almost analytic, it will also strictly contravariant almost analytic, but the converse is not necessarily true. Theorem 3.2. The condition that a vector V be strictly contravariant almost analytic in V n,is [V,X]+[V,X]+V (u i (X)) = u i ([V,X]) V (u i (X)) + u i ([V,X]) + [V,X]+[V,X] (3.5) Proof. Replacing V by V in (3.4a), we get Again, from (2.4), we have (L V F )(X) =[V,X] [V,X] (3.6a) (L V u i )(X) =V (u i (X)) u i ([V,X]) (3.6b) Adding (3.4a), (3.4b), (3.6a) & (3.6b) and using (3.1) & (3.2), we get (3.5).

8 88 R. P. Singh and S. D. Singh Corollary If vector field V be strictly contravariant almost analytic in V n, we have u i ([V,U j ]+[V,U j ]) = p i j ([V,U i]+[v,u i ]) + ([U j,v]+[u j, V ]) (3.7) Proof. By replacing X by U j in (3.5) and using (1.2) & (1.6), we get (3.7). Theorem 3.3. In V n, we have (L V u i )(X)+(L V F )(X) (L V u i )(X) (L V F )(X) = N(V,X)+(D V u i )(X) (D V u i )(X)+u i ((D X F )(V )) + u i (D X V D X V ) (3.8) Proof. Replacing V by V in (3.4b), we get (L V u i )(X) =(D V u i )(X)+u i ((D X F )(V )) + u ( D X V ) (3.9a) Operating F on both sides of (3.4a) & (3.4b), we get (L V F )(X) =[V,X] [V,X] (3.9b) and (L V u i )(X) =(D V u i )(X)+u i (D X V ) (3.9c) respectively. Adding (3.9a) & (3.6a) and subtracting (3.9b) & (3.9c) from the sum and using (2.12),we get (3.8). Theorem 3.4. In V n,ifv is strictly contravariant almost analytic, U i will contravariant almost analytic. Proof. If V is strictly contravariant almost analytic, we have (3.1a), (3.1b), (3.2a) and (3.2b). Replacing V by V in (3.2a) and using (1.1), we get 0=(L V F )(X) =(L (b 2 V +c u i (V )U i )F )(X) =b 2 (L V F )(X)+cu i (V )(L Ui F )(X) Using (3.1a) in this equation, we get (L Ui F )(X) = 0 (3.10) Similarly, replacing V by V in (3.2b) and using (1.1), we get 0=(L V u i )(X) =(L (b 2 V +c u i (V )U i )u i )(X) =b 2 (L V u i )(X)+cu i (V )(L Ui u i )(X) Using (3.1b) in this equation, we get (L Ui u i )(X) = 0 (3.11) Equations (3.11) and (3.12) prove the required statement.

9 Lie derivatives and almost analytic vector fields 89 Theorem 3.5. In V n,ifx is contravariant almost analytic vector field, we have N(X, Y )+c{u i (Y )[X, U i ]+(u i (D Y X)+(D X u i )(Y ))U i } = 0 (3.12) Proof. If X is contravariant almost analytic vector field, we have (L X F )(Y ) = 0 (3.13a) and (L X u i )(Y ) = 0 (3.13b) Replacing X by X and Y by Y alternately in (3.14a), we get Using these equations in (2.15), we get (3.13). (L X F )(Y ) = 0 (3.13c) and (L X F )(Y ) = 0 (3.13d) Theorem 3.6. In V n,ifx and Y are contravariant almost analytic vector fields, we have μ(x, Y )+p i ju j ([X, Y ]) = u i (D Y X D X Y +(D Y F ) (X) (D X F )(Y ) (3.14) Proof. (3.15) follows from (2.24) and (3.1). Theorem 3.7. In V n,ifu i is contravariant almost analytic vector field, we have (a) ν(x) =0 (b) σ(x) =0 References [1] Boothby, W. M. An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, [2] Blair, D. E. Almost contact manifolds with kliing structure tensors. Pacific J. Math. 39 (1971), [3] Hit, R. On almost complex and almost contact manifold. Revue de Faculte des Sciences de I Universite, d Istanbul 39, A (1974), 1 5. [4] Kobayasi, S., and Nomizu, K. Foudation of Differential Geometry, Vol. I, Reprint of the 1963 Original. Willely Classical Library, John Wiley and Sons, Inc., New York, 1996.

10 90 R. P. Singh and S. D. Singh [5] Mishra, R. S. On almost contact manifold. Indian J. Pure and App.Math. 5, 2 (1974), [6] Mishra, R. S. Structure on a Differentiable Manifold and their Application,. Chandrma Prakashan, Allahabad, India, [7] Sawaki, S. On almost analytic vectors in almost Hermition spaces. J. Fac. Sci., Niigata Uni. 3 (1960), [8] Yano, K., and Ako, M. Almost analytic vectors in almost complex spaces. Tohoku Math. J. 13 (1961), Received: April, 2009