Pseudo Ricci symmetric real hypersurfaces of a complex projective space are classified and it is proved that there are no pseudo Ricci symmetric real hypersurfaces of the complex projective space CPn for which the vector field ξ from the almost contact metric structure (φ, ξ, η, g) is a principal curvature vector field.

For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manif...

In the present work, we study the properties of biological networks by applying analogous notions of fundamental concepts in Riemannian geometry and optimal mass transport to discrete networks described by weighted graphs. Specifically, we employ possible generalizations of the notion of Ricci curvature on Riemannian manifold to discrete spaces in order to infer certain robustness properties of...

We define a gradient Ricci soliton to be rigid if it is a flat bundle N×ΓR k where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.

We prove that if (X, d,m) is a metric measure space with m(X) = 1 having (in a synthetic sense) Ricci curvature bounded from below by K > 0 and dimension bounded above by N ∈ [1,∞), then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any K ∈ R, N ≥ 1 and upper diameter bounds) hold, i.e. the isoper...

We prove that for non-branching metric measure spaces the local curvature condition CDloc(K, N) implies the global version of MCP(K, N). The curvature condition CD(K, N) introduced by the second author and also studied by Lott & Villani is the generalization to metric measure space of lower bounds on Ricci curvature together with upper bounds on the dimension. This paper is the following step o...

1 Introduction The uniformalization of Kähler manifold has been an important topic in geometry for long. The famous Frankel Conjecture states: Compact n-dimensional Kähler Manifolds of Positive Bisectional Curvature are Bi-holomorphic to P n C. [10] by using stable harmonic maps in the context of Kähler geometry. Actually, Mori proved the Harthshorne Conjecture: Every irreducible n-dimensional ...

Let M = M1 ×M2 be a product of complex manifolds. We prove that M cannot admit a complete Kähler metric with sectional curvature K < c < 0 and Ricci curvature Ric > d, where c and d are arbitrary constants. In particular, a product domain in Cn cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe that there are complete Kähler metrics with nega...

We study Ollivier-Ricci curvature, a discrete version of Ricci curvature, which has gained popularity over the past several years and has found applications in diverse fields. However, the Ollivier-Ricci curvature requires an optimal mass transport problem to be solved, which can be computationally expensive for large networks. In view of this, we propose two alternative measures of curvature t...

Let s ≥ 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s, s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modeled on that of a local symmetric space. They are spacelike Jordan Osserman with a Jacobi operator which is nilpotent of ord...