Scalar curvature

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)

References

[a1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1969)