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Given a Riemannian metric g, the scalar curvature Scal is defined as the trace of the Ricci curvature tensor with respect to the metric: [1] = . The scalar curvature cannot be computed directly from the Ricci curvature since the latter is a (0,2)-tensor field; the metric must be used to raise an index to obtain a (1,1)-tensor field in order to take the trace.
The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature.
Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor, of which the Ricci curvature is a natural by-product, would correspond to the full matrix of second derivatives of a function.
With this convention, the Ricci tensor is a (0,2)-tensor field defined by R jk =g il R ijkl and the scalar curvature is defined by R=g jk R jk. (Note that this is the less common sign convention for the Ricci tensor; it is more standard to define it by contracting either the first and third or the second and fourth indices, which yields a Ricci ...
It is obtained by averaging certain portions of the Riemann curvature tensor. The scalar curvature: R, the simplest measure of curvature, assigns a single scalar value to each point in a space. It is obtained by averaging the Ricci tensor. The Riemann curvature tensor can be expressed in terms of the covariant derivative.
In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold M and a smooth, real-valued function ƒ on M, construct a Riemannian metric on M whose scalar curvature equals ƒ. Due primarily to the work of J. Kazdan and F. Warner in the 1970s, this problem is well understood.
Then there exists a positive and smooth function f on M such that the Riemannian metric fg has constant scalar curvature. By computing a formula for how the scalar curvature of fg relates to that of g, this statement can be rephrased in the following form: Let (M,g) be a closed smooth Riemannian manifold.
where Δ is the Laplace-Beltrami operator (of negative spectrum), and R is the scalar curvature. This operator often makes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric. If n ≥ 3 and g is a metric and u is a smooth, positive function, then the conformal metric