Scalar curvature
of a Riemannian manifold at a point
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) |
Scalar curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scalar_curvature&oldid=48614