A note on the Lichnerowicz vanishing theorem for proper actions

TL;DR

This paper extends the Lichnerowicz vanishing theorem to non-compact spin manifolds with proper cocompact group actions, broadening its applicability beyond unimodular groups.

Contribution

It generalizes previous results by proving a vanishing theorem for a wider class of group actions on non-compact manifolds.

Findings

Proves a Lichnerowicz type vanishing theorem for non-compact spin manifolds with proper cocompact actions.

Extends Liu's result from unimodular groups to more general group actions.

Abstract

We prove a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions. This extends a previous result of Ziran Liu who proves it for the case where the acting group is unimodular.