mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on...
35 KB (5,029 words) - 23:36, 30 May 2024
contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number...
44 KB (6,432 words) - 08:44, 10 June 2024
generalized scalar curvature. As such, Schoen and Yau's approach originated in their study of Riemannian manifolds of positive scalar curvature, which is...
116 KB (10,419 words) - 02:34, 15 June 2024
number of highly influential contributions to the study of positive scalar curvature. By an elementary but novel combination of the Gauss equation, the...
31 KB (3,275 words) - 01:12, 9 April 2024
constant scalar curvature Kähler metric (cscK metric), is (as the name suggests) a Kähler metric on a complex manifold whose scalar curvature is constant...
8 KB (1,035 words) - 15:52, 30 December 2023
Riemannian geometry (section Positive scalar curvature)
positive scalar curvature. If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most...
13 KB (1,471 words) - 06:45, 2 May 2024
basis. Starting with dimension 3, scalar curvature does not describe the curvature tensor completely. Ricci curvature is a linear operator on tangent space...
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\operatorname {Ric} } and R {\displaystyle R} denote the Ricci curvature and scalar curvature of g {\displaystyle g} . The name of this object reflects the...
34 KB (5,859 words) - 15:32, 23 May 2024
a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature. The principal symbol of the map g ↦ Rm g {\displaystyle g\mapsto \operatorname...
20 KB (5,396 words) - 04:22, 9 May 2024
In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold M and a smooth...
2 KB (195 words) - 00:22, 12 August 2023