Scalar curvature: Difference between revisions
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==Definition== | ==Definition== | ||
=== | ===In terms of the Ricci curvature tensor=== | ||
The '''scalar curvature''' associated to <math>(M,g)</math> is defined as the trace of the [[Ricci curvature tensor]]. | The '''scalar curvature''' associated to <math>(M,g)</math> is defined as the trace of the [[Ricci curvature tensor]]. | ||
===In terms of the Ricci curvature=== | |||
The scalar curvature is a scalar function that associates a ''curvature'' at every point <math>p \in M<math> as follows. Consider an orthonormal basis for <math>T_p(M)</math>. Then, the scalar curvature at <math>p</math> is the sum of the Ricci curvatures for all vectors in the orthonormal basis. | |||
==Related notions== | |||
===Related metric properties=== | |||
* [[Constant scalar curvature metric]] | |||
* [[Positive scalar curvature metric]] | |||
Revision as of 19:03, 7 March 2007
This article defines a notion of curvature for a differential manifold equipped with a Riemannian metric
This article defines a scalar function on a manifold, viz a function from the manifold to real numbers. The scalar function may be intrinsic or defined in terms of some other structure/functions
Definition
In terms of the Ricci curvature tensor
The scalar curvature associated to is defined as the trace of the Ricci curvature tensor.
In terms of the Ricci curvature
The scalar curvature is a scalar function that associates a curvature at every point . Then, the scalar curvature at is the sum of the Ricci curvatures for all vectors in the orthonormal basis.