Abstract
For a pseudo-Riemannian manifold (M, g) of dimensionn≥3, we introduce a scalar curvature functionS(V) for non-degenerate subspacesV ofT pM which is a generalization of the scalar curvature, and give some characterizations of Einstein spaces in terms of this scalar curvature function. We also give a characterization for spaces of constant curvature. As an application of our results, we show that the Ricci curvature or the sectional curvature of a Lorentz manifold is constant if the scalar curvature function for non-degenerate subspaces is bounded.
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Partially supported by the grants from TGRC.
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Bahn, H., Hong, S. On the curvatures of Einstein spaces. Ann Glob Anal Geom 13, 91–98 (1995). https://doi.org/10.1007/BF00774571
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DOI: https://doi.org/10.1007/BF00774571