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...

generalized scalar curvature. As such, Schoen and Yau's approach originated in their study of Riemannian manifolds of positive scalar curvature, which is...

to the entire manifold, and many special metrics such as constant scalar curvature metrics and Kähler–Einstein metrics are constructed intrinsically using...

\operatorname {Ric} } and R {\displaystyle R} denote the Ricci curvature and scalar curvature of ⁠ g {\displaystyle g} ⁠. The name of this object reflects...

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...

basis. Starting with dimension 3, scalar curvature does not describe the curvature tensor completely. Ricci curvature is a linear operator on tangent space...

scalar curvature cannot exist on such manifolds. A particular consequence is that the torus cannot support any Riemannian metric of 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...

geometry, a constant scalar curvature Kähler metric (cscK metric) is a Kähler metric on a complex manifold whose scalar curvature is constant. A special...

number of highly influential contributions to the study of positive scalar curvature. By an elementary but novel combination of the Gauss equation, the...

a b {\displaystyle R^{ab}} is the Ricci curvature tensor and R {\displaystyle R} is the Ricci scalar curvature (obtained by taking successive traces of...

component. The Gaussian curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold, while...

is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter...

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...

}-{\frac {1}{2}}Rg_{\mu \nu },} where Rμν is the Ricci curvature tensor, and R is the scalar curvature. This is a symmetric divergenceless second-degree tensor...

In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold M and a smooth...

were directly transferable to derive Harnack inequalities for the scalar curvature along a positively-curved Ricci flow on a two-dimensional closed manifold...

potentials of a compact Kähler manifold whose critical points are constant scalar curvature Kähler metrics. The Mabuchi functional was introduced by Toshiki Mabuchi...

L ∝ R {\displaystyle L\,\propto \,R} where R is the scalar curvature, a measure of the curvature of space. Almost every theory described in this article...

where R g {\displaystyle R_{g}} is the scalar curvature of g {\displaystyle g} . That is, the Ricci curvature becomes proportional to the metric. A Riemannian...

is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is analogue of an n-sphere, with a Lorentzian metric in place of...

transformation. Other examples of Lorentz scalars are the "length" of 4-velocities (see below), or the Ricci curvature in a point in spacetime from general...

geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: Let (M,g) be a closed smooth Riemannian manifold...

the scalar curvature is n ( n − 1 ) κ . {\displaystyle n(n-1)\kappa .} In particular, any constant-curvature space is Einstein and has constant scalar curvature...

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...

K-stability is conjectured to be equivalent to the existence of constant scalar curvature Kähler metrics (cscK metrics). In 1954, Eugenio Calabi formulated a...

construct a bundle metric defined on the entire bundle. Computing the scalar curvature of this bundle metric, one finds that it is constant on each fiber:...

and the Laplace–Beltrami operator acting on spinors, in which the scalar curvature appears in a natural way. The result is significant because it provides...

subgroup conjecture (Jeremy Kahn, Vladimir Markovic, 2009) Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Zhiqin Lu, 2007) Nirenberg–Treves...

different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest...