Scalar Curvature Estimates by Parallel Alternating Torsion

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