Symmetry in complex contact manifolds
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CitationKorkmaz, B. (2017). Symmetry in complex contact manifolds. Hittite Journal of Science and Engineering, 4(2), 165-168.
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 (). There are two generalizations to contact metric mani-folds. In , contact metric manifolds satisfying the cur-vature condition (1.1) are called locally φ-symmetric. In  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 (). This gives a set of non Sasakian examples.Symmetry for complex contact metric manifolds is studied by Blair and Mihai in , . 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  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.
SourceHittite Journal of Science and Engineering