# Positive scalar curvature and the Euler class

## TL;DR

This paper generalizes the Lichnerowicz vanishing theorem by linking positive scalar curvature on spin manifolds with the Euler class of flat vector bundles, establishing a new topological constraint.

## Contribution

It extends classical results by connecting positive scalar curvature with the Euler class of flat bundles on spin manifolds.

## Key findings

- Proves a generalized vanishing theorem for flat vector bundles.
- Shows that the pairing of the A-hat genus, Euler class, and fundamental class vanishes under positive scalar curvature.
- Establishes a new topological obstruction related to flat bundles and scalar curvature.

## Abstract

We prove the following generalization of the classical Lichnerowicz vanishing theorem: if $F$ is an oriented flat vector bundle over a closed spin manifold $M$ such that $TM$ carries a metric of positive scalar curvature, then $<\widehat A(TM)e(F),[M]>=0$, where $e(F)$ is the Euler class of $F$.