# Enlargeable Length-structures and Scalar Curvatures

## TL;DR

This paper introduces enlargeable length-structures on manifolds and demonstrates they obstruct the existence of positive scalar curvature metrics, extending results to arbitrary dimensions and linking to CAT(0) geometries.

## Contribution

It defines enlargeable length-structures and proves they prevent positive scalar curvature on connected sums, generalizing previous results to all dimensions.

## Key findings

- Enlargeable length-structures obstruct positive scalar curvature.
- Closed manifolds with strongly equivalent CAT(0)-metrics are examples of enlargeable structures.
- Obstructions to positive scalar curvature are characterized by positive MV-scalar curvature.

## 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.