Garding cones and positivity of curvature operators

TL;DR

This paper investigates the relationship between Garding cones and curvature operator positivity, establishing new geometric and topological constraints on Riemannian manifolds through algebraic positivity conditions.

Contribution

It demonstrates the inclusion of the shift cone in the positivity cone and links algebraic positivity conditions to geometric and topological properties of manifolds.

Findings

Shift cone $ar{ ext{Gamma}}^{+}_2(eta)$ is contained in $ar{ ext{P}}_m$

Positivity conditions lead to vanishing theorems for Betti numbers

Characterization of spherical space forms

Abstract

This article explores the relationship between Garding cones, demonstrating that the shift cone $\overline{\Gamma}^{+}_{2}(\alpha)$ is contained in $\overline{\mathcal{P}}_{m}$. By combining these results with the study of positivity properties of curvature operators, we establish several new connections between algebraic positivity conditions and the geometry of underlying Riemannian manifolds. Our main theorems reveal how shifted cone conditions on curvature operators-both standard and of the second kind-constrain topology, including vanishing theorems for Betti numbers and characterizations of spherical space forms.