The Weighted Connection and Sectional Curvature for Manifolds With Density

Abstract

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger's finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.

Ricci Flow on Torus Bundles

“…n−1 X, and was further studied in [KWY19,Yer20]. At present, no connection is known to give Ric N f in general for any other value of N .…”

Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

“…Hence, the framework of weighted manifolds provides an extension of Riemannian geometry where many classical questions are being analyzed in recent years. In particular, comparison geometry and topological obstructions for the different weighted curvatures have been considered by many authors, see [43,44,26,3,34,45,46,28,36,47,22,23] but this list is far from exhaustive.…”

Holonomy of Manifolds with Density

“…In [WY16], Wylie and the author introduced a torsion-free affine connection whose Ricci tensor is Ric 1 f : ∇ ϕ X Y = ∇ X Y − dϕ(X)Y − dϕ(Y )X, where ϕ = f n−1 , and ∇ denotes the Levi-Civita connection. In that paper, and later in [KWY19], the connection was used for results in multiple aspects of geometry. In this paper, our focus will be on the holonomy of this connection.…”