Symmetry in complex contact manifolds

Abstract

Takahashi defined local φ-symmetry for Sasaki-an manifolds by the curvature condition that (()( , ) , )0gR Y ZWTX∇=0 for all horizontal vector fields ,,, ,X Y ZWT ([12]). There are two generalizations to contact metric mani-folds. In [2], contact metric manifolds satisfying the cur-vature condition (1.1) are called locally φ-symmetric. In [6] another definition is given. A contact metric ma-nifold is called locally φ-symmetric if characteristic reflections are local isometries. This condition leads to infinitely many curvature conditions including the abo-ve condition (1.1). Boeckx proved that ( ) ,κμ-spaces sa-tisfy this condition ([5]). This gives a set of non Sasakian examples.Symmetry for complex contact metric manifolds is studied by Blair and Mihai in [3], [4]. They defined a complex contact metric manifold to be GH-locally symmetric if the reflections in the integral submani-folds of the vertical bundle are isometries. They also proved in [4] that a complex ( ) ,κμ-space with 1κ< is GH-locally symmetric.In this paper, we will use the first generalization of local symmetry and define a complex contact metric manifold to be locally -symmetric (in order not to confuse with GH-locally symmetric) if it satisfies the curvature condition (1) and we will give a simple anddetailed proof showing that complex ( ) ,κμ-spaceswith 1κ< satisfy this condition.