Remarks on the Quadratic Orthogonal Bisectional Curvature

TL;DR

This paper explores the relationship between quadratic orthogonal bisectional curvature, combinatorics, and distance geometry, revealing new insights into curvature operators via graph energy representations.

Contribution

It establishes a novel connection between curvature concepts and graph theory, providing a new perspective on curvature operators in differential geometry.

Findings

Curvature operator can be represented as Dirichlet energy of a weighted graph.

Differences between quadratic orthogonal bisectional curvature and real bisectional curvature are clarified.

Links between curvature, combinatorics, and distance geometry are demonstrated.

Abstract

We exhibit a curious link between the Quadratic Orthogonal Bisectional Curvature, combinatorics, and distance geometry. The Weitzenb\"ock curvature operator, acting on real (1,1)--forms, is realized as the Dirichlet energy of a finite graph, weighted by a matrix of the curvature. These results also illuminate the difference in the nature of the Quadratic Orthogonal Bisectional Curvature and the Real Bisectional Curvature.