Curvature and Weitzenbock formula for spectral triples

TL;DR

This paper develops a noncommutative geometric framework for curvature tensors and the Weitzenbock formula within spectral triples, applying it to deformations of classical manifolds and analyzing their geometric invariance.

Contribution

It introduces a noncommutative curvature tensor framework and derives a general Weitzenbock formula for spectral triples, extending classical geometric concepts to noncommutative geometry.

Findings

Riemann and Ricci tensors transform naturally under $ heta$-deformation.

Connection Laplacian and scalar curvature are invariant under deformation.

Provides a noncommutative analogue of classical curvature concepts.

Abstract

Using the Levi-Civita connection on the noncommutative differential one-forms of a spectral triple $(\B,\H,\D)$, we define the full Riemann curvature tensor, the Ricci curvature tensor and scalar curvature. We give a definition of Dirac spectral triples and derive a general Weitzenbock formula for them. We apply these tools to $\theta$-deformations of compact Riemannian manifolds. We show that the Riemann and Ricci tensors transform naturally under $\theta$-deformation, whereas the connection Laplacian, Clifford representation of the curvature and the scalar curvature are all invariant under deformation.