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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.
If the scalar curvature is constant so that is cscK, then the associated holomorphy potential is a constant function, and the induced holomorphic vector field is the zero vector field. In particular on a Kähler manifold which admits no non-zero holomorphic vector fields, the only holomorphy potentials are constant functions and every extremal ...
Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions. They are particularly important in relativity theory, where they both appear on the side of Einstein's field equations that represents the geometry of spacetime (the other side of which represents the presence of matter and energy).
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.
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.
A constant scalar curvature means a general relativity gravity-like bending of spacetime that has a curvature described by a single number that is the same everywhere in spacetime in the absence of matter or energy. Negative curvature means curved hyperbolically, like a saddle surface or the Gabriel's Horn surface, similar to that of a trumpet ...
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.
where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation. These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880. [2]