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In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. It assigns to each point on a Riemannian manifold a single real number characterizing the intrinsic curvature of the manifold at that point.
In two dimensions the scalar curvature completely characterizes the curvature of a Riemannian manifold. In dimensions ≥ 3, however, more information is needed. See curvature of Riemannian manifolds for a complete discussion.
The scalar curvature is defined as the trace of the Ricci curvature with respect to the metric:
- <math>S = \mbox{tr}_g\,Ric<math>
The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first contract with the metric to obtain a (1,1)-valent tensor in order take the trace (see musical isomorphisms). In terms of local coordinates one can write
- <math>S = {R^i}_i = g^{ij}R_{ij}<math>
where
- <math>Ric = R_{ij}\,dx^i\otimes dx^j<math>
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