2024/04/25 by Gregor Weingart, Weingart, Gregor
Mathematics · #53-08 (Primary) 53C25 #53E20 (Secondary) #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2404.16355
openalex publication_date 2024/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we develop a graphical calculus for stable invariants of Riemannian manifolds akin to the graphical calculus for Rozansky-Witten invariants for hyperkähler 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-Gauß-Bonnet and the normalized moment polynomials calculating the moments of sectional curvature considered as a random variable on the Graßmannian 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 L2--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.