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...

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

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

constant scalar curvature Kähler metric (cscK metric), is (as the name suggests) a Kähler metric on a complex manifold whose scalar curvature is constant...

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...

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

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...