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Scalar curvature

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of a Riemannian manifold at a point

The trace of the Ricci tensor with respect to the metric tensor . The scalar curvature is connected with the Ricci curvature and the sectional curvature by the formulas

where is an orthonormal basis of the tangent space. In the equivalent Einstein notation, these equations have the form

where and are the components of the Ricci tensor and the curvature tensor, respectively, and the are the contravariant components of the metric tensor.

References

[1] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)


Comments

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1969)
How to Cite This Entry:
Scalar curvature. L.A. Sidorov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Scalar_curvature&oldid=12835
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098