We introduce the pseudo-Einstein structure on real hypersurfaces in a Kählerian manifold, namely, the Ricci curvature tensor for the generalized Tanaka-Webster connection (restricted) on the Levi subbundle D is proportional to the Levi form. In particular, we give a classification of pseudo-Einstein Hopf-hypersurfaces in a non-flat complex space form.
Abstract. We prove that a semi-symmetric 3-dimensional gradient Ricci soliton is locally isometric to a space form S 3 , H 3 , R 3 (Gaussian soliton); or a product space R × S 2 , R × H 2 , where the potential function depends only on the nullity.
We prove that a contact strongly pseudo-convex CR (Cauchy–Riemann) manifold M2n+1, n≥2, is locally pseudo-Hermitian symmetric and satisfies ∇ξh=μhϕ, μ∈R, if and only if M is either a Sasakian locally ϕ-symmetric space or a non-Sasakian (k,μ)-space. When n=1, we prove a classification theorem of contact strongly pseudo-convex CR manifolds with pseudo-Hermitian symmetry.
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