Sectional curvature and Weitzenböck formulae

Abstract

We establish a new algebraic characterization of sectional curvature bounds $\sec \ge k$ and $\sec \le k$ using only curvature terms in the Weitzenböck 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 $\sec \ge 0$, obtaining new insights into the Hopf Conjecture, without any symmetry assumptions.

LaTeX2e, 25 pages, final version. To appear in Indiana Univ. Math. J

References in corpus (3)

• Examples of Riemannian Manifolds with non-negative sectional curvature
• Killing and Conformal Killing tensors
• ${\mathsf{S}}^1$-actions on $4$-manifolds and fixed point homogeneous manifolds of nonnegative curvature

Cited by in corpus (5)

• New Curvature Conditions for the Bochner Technique
• Extremality and rigidity for scalar curvature in dimension four
• Geometry in the large of the kernel of Lichnerowicz Laplacians and its applications
• Curvature operators and rational cobordism
• Geography of pinched four-manifolds