Algebraic Properties of Riemannian Manifolds

Abstract

Algebraic properties are explored for the curvature tensors of Riemannian manifolds, using the irreducible decomposition of curvature tensors. Our method provides a powerful tool to analyze the irreducible basis as well as an algorithm to determine the linear dependence of arbitrary Riemann polynomials. We completely specify 13 independent basis elements for the quartic scalars and explicitly find 13 linear relations among 26 scalar invariants. Our method provides several completely new results, including some clues to identify 23 independent basis elements from 90 quintic scalars, that are difficult to find otherwise.

arXiv commentA few typos corrected; 40 pages (4 appendices: 16 pages). To appear in General Relativity and Gravitation

References in corpus (8)

• The Invar Tensor Package
• Spacetimes characterized by their scalar curvature invariants
• Dimensionally Dependent Tensor Identities by Double Antisymmetrisation
• Yang-Mills Instantons from Gravitational Instantons
• An Efficient Representation of Euclidean Gravity I
• Einstein Manifolds As Yang-Mills Instantons
• Anatomy of Einstein Manifolds
• Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons and Mirror Symmetry

Cited by in corpus (4)

• Trace Anomaly of Weyl Fermions via the Path Integral
• Generalization of Instanton-Induced Inflation and Dynamical Compactification
• Explicit Construction of Hermitian Yang-Mills Instantons on Coset Manifolds
• Einstein Structure of Four-Manifolds