Algebraic properties of Riemannian manifolds

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.