NON-EXISTENCE OF LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KAEHLER MANIFOLDS ADMITTING NON-METRIC π-CONNECTIONS

Abstract

Abstract. In this paper, we study two types 1-lightlike submanifoldsM, so called lightlike hypersurface and half lightlike submanifold, of anindefinite Kaehler manifold M¯ admitting non-metric π-connection. Weprove that there exist no such two types 1-lightlike submanifolds of anindefinite Kaehler manifold M¯ admitting non-metric π-connections. 1. IntroductionA linear connection ∇¯ on a semi-Riemannian manifold (M,¯ ¯g) is called anon-metric π-connection if, for any vector fields X, Y and Z on M¯, it satisfies(1.1) (∇¯ X g¯)(Y,Z) = −π(Y )¯g(X,Z)−π(Z)¯g(X,Y ),where π is a 1-form, associated with a non-vanishing smooth vector field ζ onM¯ by π(X) = ¯g(X,ζ). We say that ζ is the characteristic vector field of M¯.Two special cases are important for both the mathematical study and theapplications to physics: (1) A non-metric π-connection ∇¯ on M¯ is called asemi-symmetric non-metric connection if its torsion tensor T¯ satisfiesT¯(X,Y ) = π(Y )X −π(X)Y.The notion of semi-symmetric non-metric connections on a Riemannian man-ifold was introduced by Ageshe and Chafle [1] and later studied by many au-thors. The lightlike version of Riemannian manifolds with semi-symmetricnon-metric connections have been studied by some authors [15, 16, 17, 18, 23].(2) A non-metric π-connection ∇¯ on M¯ is called a quarter-symmetric non-connection if its torsion tensor T¯ satisfiesT¯(X,Y) = π(Y )φX −π(X)φY,where φ is a (1,1)-type tensor field. In particular, if φX = X, then the quarter-symmetric connection reduces to the semi-symmetric connection [8]. Thus the