Pseudo-Hermitian symmetries
Let \((M,\eta, J, g)\) be a strictly pseudo-convex CR manifold with a contact form \(\eta\), complex structure \(J\) on \(D = ker (\eta)\) and the Riemannian metric \[ g := \eta \otimes \eta + d \eta(J \cdot, \cdot)| _{D}. \] The authors say that this manifold is a Tanaka-Webster parallel space if the Tanaka-Webster connection \(\tilde \nabla\) has \(\tilde \nabla\)-parallel curvature \(\tilde R\) and torsion \(\tilde T\) tensors. The manifold \(M\) is called a weakly locally pseudo-Hermitian symmetric space if \[ g(\tilde \nabla_X \tilde R(Y,Z)U,V) =0 \] for all \(X,Y,Z,U,V \in D\). The authors give a description of Tanaka-Webster parallel spaces and weakly locally pseudo-Hermitian spaces. In particular, they prove that a Sasakian manifold is a locally pseudo-Hermitian symmetric space if and only if it is a locally Sasakian \(\varphi\)-symmetric space in the sense of Takahashi.
scientific article; zbMATH DE number 5351776
scientific article