Deformation and rigidity results for the 2k-Ricci tensor and the 2k-Gauss-Bonnet curvature

TL;DR

This paper investigates deformation and rigidity properties of closed Riemannian manifolds with constant 2k-Ricci tensor or 2k-Gauss-Bonnet curvature, using explicit linearization formulas derived from double forms.

Contribution

It provides new deformation and rigidity results for classes of manifolds with specific curvature invariants, expanding understanding of their geometric structure.

Findings

Results apply to all non-flat space forms.

Explicit linearization formulas for curvature invariants.

Established rigidity under certain conditions.

Abstract

We present several deformation and rigidity results within the classes of closed Riemannian manifolds which either are $2k$-Einstein (in the sense that their $2k$-Ricci tensor is constant) or have constant $2k$-Gauss-Bonnet curvature. The results hold for a family of manifolds containing all non-flat space forms and the main ingredients in the proofs are explicit formulae for the linearizations of the above invariants obtained by means of the formalism of double forms.