Geometry and symmetry on Sasakian manifolds
Let M denote a Sasakian manifold and let \(\phi\) be the tensor field of type (1,1) with the property \(\phi^ 2=-I+\eta \otimes \xi\) where \(\eta\) denotes the contact form and \(\xi\) is the characteristic vector field with \(\eta (\xi)=1\). A geodesic \(\gamma\) on M is called a \(\phi\)-geodesic if \(\eta\) (\({\dot \gamma}\))\(=0\). M is locally \(\phi\)-symmetric (in the sense of Takahashi) if \(\phi^ 2(\nabla_ VR)(X,Y)Z=0\) for all vector fields V,X,Y,Z orthogonal to \(\xi\). A plane section of \(T_ pM\) is called a \(\phi\)-section if it is spanned by vectors X and \(\phi\) X orthogonal to \(\xi\). The sectional curvature of a \(\phi\)-section is called a \(\phi\)-sectional curvature. M is a Sasakian space form if M is of constant \(\phi\)-sectional curvature. In the paper the following main results are proved: M is locally \(\phi\)- symmetric iff the volume density function \(\theta_{\sigma}\) has antipodal symmetry along \(\phi\)-geodesics orthogonal to a \(\phi\)-geodesic \(\sigma\) for any \(\sigma\). M is a Sasakian space form iff the local symmetries with respect to all \(\phi\)-geodesics are volume-preserving provided that M is connected. M has constant curvature 1 iff M is harmonic with respect to each \(\phi\)-geodesic. In the case dim(M)\(\geq 5\) this condition is satisfied iff the local symmetries with respect to all \(\phi\)-geodesics are isometries. If M is a 3-dimensional Sasakian space form then any local symmetry with respect to any \(\phi\)-geodesic is an isometry.
scientific article; zbMATH DE number 4075996
scientific article