Scalar curvature
From Encyclopedia of Mathematics
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
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where is an orthonormal basis of the tangent space. In the equivalent Einstein notation, these equations have the form
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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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scalar_curvature&oldid=12835
Scalar curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scalar_curvature&oldid=12835
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article