Positive weighted sectional curvature

Abstract

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field that gives them positive weighted curvature. On the other hand, we generalize a number of the foundational results for compact manifolds with positive sectional curvature to positive weighted curvature. In particular, we prove generalizations of Weinstein's theorem, O'Neill's formula for submersions, Frankel's theorem, and Wilking's connectedness lemma. As applications of these results, we recover weighted versions of topological classification results of Grove-Searle and Wilking for manifolds of high symmetry rank and positive curvature.

References in corpus (9)

• Examples of Riemannian Manifolds with non-negative sectional curvature
• Nonnegatively and Positively curved manifolds
• The classification of compact simply connected biquotients in dimension 6 and 7
• Tits Geometry and Positive Curvature
• Topological obstructions to fatness
• Nonnegatively curved 5-manifolds with non-abelian symmetry
• Positively curved metrics on symmetric spaces with large symmetry rank
• $(K,N)$-convexity and the curvature-dimension condition for negative $N$
• Fundamental groups of manifolds with positive sectional curvature and torus symmetry

Cited by in corpus (3)

• Some curvature pinching results for Riemannian manifolds with density
• Sectional curvature for Riemannian manifolds with density
• Compact 4-Dimensional Spin Gradient $m$-quasi-Einstein Manifolds Satisfy the Hitchin-Thorpe Inequality when $m\ge 1$