Scalar Curvature Estimates by Parallel Alternating Torsion

TL;DR

This paper extends scalar curvature comparison results to manifolds with special metric connections, linking topological invariants to holonomy, especially for quotients of compact Lie groups with homogeneous metrics.

Contribution

It generalizes Llarull's scalar curvature comparison to manifolds with parallel alternating torsion and nonnegative curvature operator, and relates topological invariants to holonomy.

Findings

Scalar curvature comparison extended to manifolds with parallel alternating torsion.

Topological invariants like Euler number and signature are determined by holonomy.

Results apply to quotients of compact Lie groups with normal homogeneous metrics.

Abstract

We generalize Llarull's scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on 2-vectors. As a byproduct, we show that Euler number and signature of such manifolds are determined by their global holonomy representation. Our result holds in particular for all quotients of compact Lie groups of equal rank, equipped with a normal homogeneous metric. We also correct a mistake in the treatment of odd-dimensional spaces in arXiv:math/0010199 and arXiv:0705.0500