Enlargeable Length-structures and Scalar Curvatures

Abstract

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed $n$-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive $MV$-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

Change the title and to appear in Annals of Global Analysis and Geometry

References in corpus (3)

• Enlargeable metrics on nonspin manifolds
• Curvature-Dimension Condition Meets Gromov's $n$-Volumic Scalar Curvature
• Contracting Maps and Scalar Curvature

Cited by in corpus (1)

• Sphere Theorems with and without Smoothing