Publications of Gray, Alfred

• Invariants of Curvature Operators of Four-Dimensional Riemannian Manifolds

  Gray A.

  Published in 1971

  In the theory of Riemannian manifolds the study of the 4-dimensional case is especially important. On the one hand, 4-dimensional Riemannian manifolds are simpler than higher even-dimensional manifolds, but at the same time they are significantly more complicated than 2-dimensional manifolds.

  There are three well-known topological invariants that one can assocoate with various classes of compact Riemannian manifolds. These are the Euler characteristic χ(M), the Hirzebruch index τ(M), and the arithmetic genus α(M). Each of these may be expressed as an integral over the Riemannian manifold M of a certain differential form derived from the curvature operator.

  The purpose of this paper is to investigate the relationship between χ(M), τ(M), and α(M), and the decomposition of the space of curvature operators at a point of M given by Singer and Thorpe.

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