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...

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...

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...

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

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

positive scalar curvature. If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most...

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

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

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...

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...

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...

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

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

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...

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors)...

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...

general relativity, curvature invariants are a set of scalars formed from the Riemann, Weyl and Ricci tensors — which represent curvature, hence the name...

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

is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian[further explanation needed] analogue of an n-sphere...

{\displaystyle R_{\mu \nu }} is the Ricci curvature tensor, and R {\displaystyle R} is the scalar curvature. This is a symmetric second-degree tensor...

metric, R is the Riemann tensor, Ric is the Ricci tensor, s is the scalar curvature, and h   ∧ ◯   k {\displaystyle h{~\wedge \!\!\!\!\!\!\!\!\;\bigcirc...

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

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

Kähler–Einstein metrics of zero scalar curvature on compact complex manifolds. The case of nonzero scalar curvature does not follow as a special case...

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...

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...

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

three dimensions of closed manifolds which admit metrics of positive scalar curvature. His third preprint (or alternatively Colding and Minicozzi's work)...

4 x {\displaystyle {\sqrt {-g}}\,\mathrm {d} ^{4}x} and the Ricci scalar curvature R {\displaystyle R} . The scale factor κ {\displaystyle \kappa } is...