Geometric formality and non-negative scalar curvature

Mass and 3-metrics of Non-negative Scalar Curvature

Physicists believe, with some justification, that there should be a correspondence between familiar properties of Newtonian gravity and properties of solutions of the Einstein equations. The Positive Mass Theorem (PMT), first proved over twenty years ago [45, 53], is a remarkable testament to this faith. However, fundamental mathematical questions concerning mass in general relativity remain, a...

Rigidity in Non-negative Curvature

In this paper we will show that any complete manifold of nonnegative curvature has a flat soul provided it has curvature going to zero at infinity. We also show some similar results about manifolds with bounded curvature at infinity. To establish these theorems we will prove some rigidity results for Riemannian submersions, eg., any Riemannian submersion with complete flat total space and compa...

Dirac Operators on Hypersurfaces of Manifolds with Negative Scalar Curvature

We give a sharp extrinsic lower bound for the first eigenvalues of the intrinsic Dirac operator of certain hypersurfaces bounding a compact domain in a spin manifold of negative scalar curvature. Limiting-cases are characterized by the existence, on the domain, of imaginary Killing spinors. Some geometrical applications, as an Alexandrov type theorem, are given. Mathematics Subject Classificati...

Geometric Realizations of Curvature Models by Manifolds with Constant Scalar Curvature

We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyperpseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and ⋆-scalar curvature.

Positive Scalar Curvature and Surgery