A Graphical Calculus for Stable Curvature Invariants

TL;DR

This paper introduces a graphical calculus for stable curvature invariants of Riemannian manifolds, enabling explicit calculations of classical polynomials and deriving new curvature identities for Einstein manifolds.

Contribution

It develops a novel graphical calculus for stable invariants, linking graph theory with curvature polynomials and providing explicit formulas and identities.

Findings

Derived a curvature identity involving Euler characteristic and curvature moments

Explicitly described Pfaffian and moment polynomials using the calculus

Provided a computer algebra implementation for practical calculations

Abstract

In this article we develop a graphical calculus for stable invariants of Riemannian manifolds akin to the graphical calculus for Rozansky-Witten invariants for hyperk\"ahler manifolds; based on interpreting trivalent graphs with colored edges as stably invariant polynomials on the space of algebraic curvature tensors. In this graphical calculus we describe explicitly the Pfaffian polynomials central to the Theorem of Chern-Gau{\ss}-Bonnet and the normalized moment polynomials calculating the moments of sectional curvature considered as a random variable on the Gra{\ss}mannian of planes. Eventually we illustrate the power of this graphical calculus by deriving a curvature identity for compact Einstein manifolds of dimensions greater than 2 involving the Euler characteristic, the third moment of sectional curvature and the $L^2$--norm of the covariant derivative of the curvature tensor. A model implementation of this calculus for the computer algebra system Maxima is available for download under http://www.matcuer.unam.mx/~gw/CurvGraphs.mac.