Discrete Bakry-\'Emery curvature tensors and matrices of connection graphs

TL;DR

This paper reformulates Bakry-Émery curvature for connection graphs using eigenvalues of curvature matrices, extending previous work and addressing unique behaviors in connection graph structures.

Contribution

It introduces a new curvature tensor framework for connection graphs, extending prior studies and analyzing curvature in product and unbalanced structures.

Findings

Curvature matrices are unitarily equivalent and represent a new tensor.

Constant functions are generally not eigenfunctions of the connection Laplacian.

Results extend to Cartesian products and reveal behaviors in unbalanced structures.

Abstract

Liu, M\"unch, and Peyerimhoff introduced the notion of Bakry-\'Emery curvature for connection graphs as a means to derive Buser-type bounds on the eigenvalues of connection Laplacians. In this work, we present a reformulation of the Bakry-'Emery curvature at a vertex within a connection graph. Our approach expresses this curvature through the smallest eigenvalue of a set of unitarily equivalent curvature matrices. We interpret these matrices as representations of a newly defined curvature tensor, each corresponding to a different orthonormal basis of the vertex's tangent space. This framework significantly extends earlier studies by Cushing et al. and Siconolfi on curvature matrices of standard graphs. It is important to note that the Bakry-\'Emery curvature in connection graphs can behave very differently from that in the underlying graphs. For instance, constant functions generally fail to serve as eigenfunctions of the connection Laplacian, which poses a substantial challenge when attempting to generalize results from standard graphs to connection graphs. We address this issue by employing the Schur complement, applied twice using pseudoinverses. Additionally, we investigate the Bakry-\'Emery curvature in Cartesian products of connection graphs, extending and strengthening the earlier findings of Liu, M\"unch, and Peyerimhoff. While our results for vertices with locally balanced structures encompass previous work, we also shed light on intriguing behaviors that arise in locally unbalanced connection structures.