Extending tensors on polar manifolds

Let $M$ be a Riemannian manifold with a polar action by the Lie group $G$, with section $\Sigma\subset M$ and generalized Weyl group $W$. We show that restriction to $\Sigma$ is a surjective map from the set of smooth $G$-invariant tensors on $M$ onto the set of smooth $W$-invariant tensors on $\Sigma$. Moreover, we show that every smooth $W$-invariant Riemannian metric on $\Sigma$ can be extended to a smooth $G$-invariant Riemannian metric on $M$ with respect to which the $G$-action remains polar with the same section $\Sigma$.Comment: arXiv admin note: text overlap with arXiv:1205.476

Sectional curvature and Weitzenb\"ock formulae

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq k$ using only curvature terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing a symmetric analogue of the Kulkarni-Nomizu product, we provide a simple formula for such curvature terms. We also give an application of the Bochner technique to closed $4$-manifolds with indefinite intersection form and $\sec>0$ or $\sec\geq0$, obtaining new insights into the Hopf Conjecture, without any symmetry assumptions.Comment: LaTeX2e, 25 pages, final version. To appear in Indiana Univ. Math.

Strongly positive curvature

We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and stems from the work of Thorpe in the 1970s. We prove that this condition is preserved under Riemannian submersions and Cheeger deformations, and that most compact homogeneous spaces with positive sectional curvature satisfy it.Comment: LaTeX2e, 26 page