Scalar curvature Guide, Meaning , Facts, Information and Description

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:

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

where

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