In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space formMm(c), c 6= 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [17, 21] for a priori estimates of the corresponding Monge-Ampère equation. It is expressed by a socalled cost-sectional curvature being non-negative. W...

We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to this theorem.

Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature.

Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, while positive curvature suggests stability. In the first part of the paper we s...

Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, while positive curvature suggests stability. In the first part of the paper we s...

The study of holomorphic and isometric immersions of a Kähler manifold (M,g) into a Kähler manifold (N,G) started with Calabi. In his famous paper [3] he considered the case when the ambient space (N,G) is a complex space form, i.e. its holomorphic sectional curvature KN is constant. There are three types of complex space forms : flat, hyperbolic or elliptic according as the holomorphic section...

In this paper, we extend the general maximum principle in [NT3] to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional curvature of dimension greater than three such that the Ricci flow does not preserve the nonnegativity of t...

We present a compared analysis of some properties of indefinite almost S-manifolds and indefinite S-manifolds. We give some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields. We study the sectional and φ-sectional curvature of indefinite almost S-manifolds and state an expression of the curvature tensor field for the indefinite S-space forms. We an...