SOME GENERALIZATIONS OF F-CONNECTIONS ON DIFFERENTIABLE MANIFOLDS

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1 SOME GENERALIZATIONS O -CONNECTIONS ON DIERENTIABLE MANIOLDS DUMITRU, Dan aculty of Mathematics-Informatics Spiru Haret University dumitru984@yahoo.com Abstract In this article we generalize the notion of -connection, where is an almost product structure on a manifold we give some characterizations of the connections compatible with those structures. Key words : (ap)-structure, -connection AMS classification: 58A5. Introduction Differentiable manifolds with almost product structures (ap)-structures, together with other important structures were investigated by G. Walker [], Wilmore [], Yano [6]. New properties of these structures were established by Legr [3] Hsu [4]. In this paper we generalize the notion of -connection for a connection on a manifold, where is an (ap)-structure, by generalizing the (ap)-structures. We also establish some properties of these connections. Let M be a differentiable manifold of dimension n TM p the tangent space to M in the point p with p M. We denote with Γ(TM) the tangent bundle of M, r with (M) the ring of real functions on M, with Ds ( M ) the (M) module of (r, s)- tensor fields, with C(M) the set of connections on M, with D ( M ) the (M) module of sections of M. Let be an almost product (ap)-structure on M, that means is an (, )- tensor field defined on M satisfying = I, where I is the identity. Setting: Ψ ( ) = ( + o o ) () 5

2 χ ( τ) = ( τ + oτ o ), where τ is an (, )-tensor field on M, D ( M ) it follows ( ) C( M) Ψ, χ ( τ ) D ( M ) () Ψ = Ψ, χ = χ, Ψ ( + τ ) = Ψ ( ) χ ( τ ). (3) + Hence ψ is the (M)-affine projector on C(M), associated to the (M)- linear projector χ on ( ) D M. Definition. A connection on M is called compatible with the (ap)- structure (or is an -connection) if it satisfies =, where Y = Y Y ( )( ) for all, Y Γ(TM). ( ) ( ) We will denote by C (M) the set of all connections compatible with. rom the expression of Ψ ( ) it follows that Ψ ( ) ( ) =, D ( M ), i.e. the image of any connection by the projector Ψ is an -connection. Conversely, if = it follows that o o = so Ψ ( ) =, D ( M) i.e. Im Ψ. Thus we have: Theorem. The set C ( M ) of the connections compatible with the (ap)- structure is the affine submodule of C(M) which is the image of the affine projector, i.e. Ψ = C M Let Ψ ( ) Im be a fixed connection on M. Since Ψ = C ( M), then for each Im connection C ( M) there exists C( M) such that =Ψ ( ) [5], there exists τ D ( M ) such that τ Then it follows:. But from = + therefore, ( τ ) =Ψ +. 6

3 Theorem. The set C ( M ) of the connections compatible with the (ap)- structure is given by ( ) χ ( τ ) =Ψ +. (4) where is a fixed connection on M τ is an arbitrary (, )-tensor field on M.. Main results 3 Let us consider an (, )-tensor field on M, such that = I. We will try to construct Ψ such that Ψ is a connection on M a similar result to the Theorem holds. or that let G = I. We have that G is also an (, )-tensor field on M ( ) 4 G I I 4 4 I = = + =. (5) So G is an (ap)-structure on M. Hence, if we set Ψ ( ) = ( + Go og ) we obtain that ( ) ( ) tensor field Ψ C M. Definition. A connection on M is called compatible with the (, )- (or is Theorem 3. The set ( ) -an connection) if it satisfies =. C M of the connections compatible with the (, )- tensor field is the affine submodule of C(M) which is the image of the affine Im Ψ = C M. projector Ψ ( ), i.e. ( ) ( ) Proof. irst of all, we will prove that Ψ takes any connection on M into an -connection. We observe that G = G = o o. Let be a connection on M. We will compute ( ) ( ) Ψ =Ψ o oψ. We have, consecutively, 7

4 ( ) ( Y) ( Y) G( G( ( Y ) )) Ψ o =Ψ = + = = ( ) ( ) ( ) ( ) Y G = Y I ( Y) = = ( ) ( ) ( ) Y Y + Y = Y ( ) ( ( )), o Ψ = ( Ψ ( ) Y) = ( Y) + ( G( ( Y) )) = = ( ) ( ( )) ( ) ( ( Y G Y Y Y ( Y) )) = = = ( ) ( ) ( ( )) ( ( )) Y Y Y + = Y. Therefore, ( ) ( ) projector Ψ = then the image of any connection by the Ψ is an -connection. Now, let be an -connection. Then Ψ ( ) ( ) ( ) ( = + Go og = + I I ) = o o = 4 + o o + o o = = + = o o. Hence ( ) C M = Im Ψ. n Let us consider an (, )-tensor field on M such that + = n I, where n is at least. We will try to generalize the construction of Ψ such that Ψ is a connection on M a similar result to the Theorem 3 holds. or that, let H = I n. We have that H is also an (, )-tensor field on M 8

5 ( n ) So H is an (ap)-structure on M. n 4 H I I 4 4 n I = = + =. (6) Thus, if we set Ψ ( ) = ( + H o oh ) we obtain that ( ) ( ) Ψ C M. Definition 3. A connection on M is called compatible with the (,) tensor field n (or is an n n -connection) if it satisfies =. Theorem 4. The set n ( ) C M of the connections compatible with the (,)- n tensor field is the affine submodule of C(M) which is the image of the affine Ψ = C M. projector Ψ ( ), i.e. ( ) ( ) n Proof. irst of all, we will prove that Ψ takes any connection on M into an n - connection. We observe that: Let be a connection on M. We will compute ( ) ( ) n n Ψ n =Ψ o o Ψ. We have n n H = H = n o o. ( ( )) n ( ) n ( ) n( ) n Ψ o =Ψ Y = Y + H H ( Y) = n( ) ( n ) n( ) ( n ) n = Y H = Y I ( Y) = n n n n n n = ( Y) ( Y) + ( ( Y) ) = ( ( Y) ). n n n n o Ψ = ( Ψ ( ) Y) = ( Y) + ( H( ( Y) )) = 9

6 n( ) n( ) n n( ) n n = Y Y + Y = Y n So ( ) ( ) ( ) ( ( )) Ψ = then the image of any connection by the projector Ψ is an n -connection. Now, let be an n -connection. Then ( ) ( ) ( n ) ( n Ψ = + H o oh = I I ) = o o. Thus n ( ) = + + = n n n n 4 o o o o C M = Im Ψ. 4 n 4 4n = + = o o. Remark. Once we have defined an almost product structure,, on a differentiable manifold, M, then M carries two globally complementary distributions D D. We denote with V V respectively the projection tensor fields corresponding to the two distributions D D. We say that is a vector field which belongs to the distribution D if for every point p in M, we p p, where Dp( M) V( TpM) have D ( M) =. Let be a linear connection on the manifold M. We say that the distribution D is parallel with respect to the connection if for every vector field which belongs to the distribution D, the vector field Y belongs to the distribution D for every vector field Y of the manifold M. We can define on the manifold M two linear connections by: ( ( )) ( ( )) QY= V V Y + V V Y, (7) ( ( ) ( )) ( ) ( ) ( ) [, ] [, ] LY= V V Y + V V Y + VVVY + V VVY. (8) V V

7 Proposition. The distributions of the almost product structure are both parallel with respect to the connections Q L, for every connection. Proof. Let Y be a vector field which belongs to the distribution D of the V Y = hence almost product structure. Then we have ( ) ( ) ( ) V Q Y =, V L Y =, for every vector field on the manifold M. Proposition. The connection Q is equal to the connection if only if the distributions of the almost product structure are parallel with respect to the connection. Proof. If the connections Q are equal then we get that ( ( )) ( ( )) V V Y + V V Y =. (9) By the properties of the projectors it follows that V( V ( Y) ) V ( V( Y) ) =. = Hence, the distributions of are parallel with respect to the connection. The converse can be verified immediately. Theorem 5. If one of the connections Q or L is symmetric, then the almost product structure is integrable. Proof. The almost product structure is integrable if the distributions D D are involutive, that is if V ( V, VY) = V( V, VY ) = for any two vector fields, Y on the manifold M. If the connection Q is symmetric then we QY Q= Y,. have that [ ] Thus the relation V ( V, VY) Y = is equivalent to the condition This condition is verified by the Proposition. V Q VY V Q V =. () Analog if L is symmetric, using the relation V( V, VY) Y VY =, the result is proved

8 REERENCES. Walker G. A., Connections for Parallel Distribution in the Large, Quart. J. Math. (Oxford), (), 6: 3-8, Wilmore T. J., Parallel Distribution on Manifolds, Proc. London. Math. Soc. (3) 6: 9-4, Legr J., Étude d'une généralisation des structures presque-complexes sur les variétés différentiables, Rend. Circ. Mat. Palermo, 7: , Hsu, C.J., On Some Properties of π-strucutres on Differentiable Manifolds, Tohoku Math. J. : , Cruceanu, V., Connections compatibles avec certaines structures sur un fibre vectoriel banachique, Czechoslovak. Math. J. 4 (99): 6-4, Yano K., Differential Geometry on Complex Almost Complex Spaces, Pergamon Press, 956.