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Scalar curvature

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of a Riemannian manifold at a point $ p $

The trace of the Ricci tensor with respect to the metric tensor $ g $. The scalar curvature $ s ( p) $ is connected with the Ricci curvature $ r $ and the sectional curvature $ k $ by the formulas

$$ s ( p) = \ \sum _ {i = 1 } ^ { n } r ( e _ {i} ) = \ \sum _ {i, j = 1 } ^ { n } k ( e _ {i} , e _ {j} ), $$

where $ e _ {1} \dots e _ {n} $ is an orthonormal basis of the tangent space. In the equivalent Einstein notation, these equations have the form

$$ s ( p) = g ^ {ij} R _ {ij} = \ g ^ {ij} g ^ {kl} R _ {kijl} , $$

where $ R _ {ij} $ and $ R _ {kijl} $ are the components of the Ricci tensor and the curvature tensor, respectively, and the $ g ^ {ij} $ 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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scalar_curvature&oldid=48614
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article