Garding cones and positivity of curvature operators

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.