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On Ricci curvature of totally real submanifolds in a quaternion projective space

Ximin Liu — 2002

Archivum Mathematicum

Let M n be a Riemannian n -manifold. Denote by S ( p ) and Ric ¯ ( p ) the Ricci tensor and the maximum Ricci curvature on M n , respectively. In this paper we prove that every totally real submanifolds of a quaternion projective space Q P m ( c ) satisfies S ( ( n - 1 ) c + n 2 4 H 2 ) g , where H 2 and g are the square mean curvature function and metric tensor on M n , respectively. The equality holds identically if and only if either M n is totally geodesic submanifold or n = 2 and M n is totally umbilical submanifold. Also we show that if a Lagrangian submanifold of...

Riemannian semisymmetric almost Kenmotsu manifolds and nullity distributions

Yaning WangXimin Liu — 2014

Annales Polonici Mathematici

We consider an almost Kenmotsu manifold M 2 n + 1 with the characteristic vector field ξ belonging to the (k,μ)’-nullity distribution and h’ ≠ 0 and we prove that M 2 n + 1 is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, provided that M 2 n + 1 is ξ-Riemannian-semisymmetric. Moreover, if M 2 n + 1 is a ξ-Riemannian-semisymmetric almost Kenmotsu manifold such that ξ belongs to the (k,μ)-nullity distribution, we prove that M 2 n + 1 is...

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