In the 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 a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.
The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of the key terms in the Einstein field equations. Furthermore, this scalar curvature is the Lagrangian density for the Einstein–Hilbert action, the Euler–Lagrange equations of which are the Einstein field equations in vacuum.
The geometry of Riemannian metrics with positive scalar curvature has been widely studied. On noncompact spaces, this is the context of the positive mass theorem proved by Richard Schoen and Shing-Tung Yau in the 1970s, and reproved soon after by Edward Witten with different techniques. Schoen and Yau, and independently Mikhael Gromov and Blaine Lawson, developed a number of fundamental results on the topology of closed manifolds supporting metrics of positive scalar curvature. In combination with their results, Grigori Perelman's construction of Ricci flow with surgery in 2003 provided a complete characterization of these topologies in the three-dimensional case.
mathematical field of Riemannian geometry, the scalarcurvature (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...
number of highly influential contributions to the study of positive scalarcurvature. By an elementary but novel combination of the Gauss equation, the...
\operatorname {Ric} } and R {\displaystyle R} denote the Ricci curvature and scalarcurvature of g {\displaystyle g} . The name of this object reflects the...
positive scalarcurvature. If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalarcurvature is at most...
a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalarcurvature. 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...
basis. Starting with dimension 3, scalarcurvature does not describe the curvature tensor completely. Ricci curvature is a linear operator on tangent space...
with the sectional curvature of the surface. It is also exactly half the scalarcurvature of the 2-manifold, while the Ricci curvature tensor of the surface...
{\displaystyle R_{\mu \nu }} is the Ricci curvature tensor, and R {\displaystyle R} is the scalarcurvature. This is a symmetric second-degree tensor...
a b {\displaystyle R^{ab}} is the Ricci curvature tensor and R {\displaystyle R} is the Ricci scalarcurvature (obtained by taking successive traces of...
In Riemannian geometry, a branch of mathematics, the prescribed scalarcurvature problem is as follows: given a closed, smooth manifold M and a smooth...
potentials of a compact Kähler manifold whose critical points are constant scalarcurvature Kähler metrics. The Mabuchi functional was introduced by Toshiki Mabuchi...
the scalarcurvature is n ( n − 1 ) κ . {\displaystyle n(n-1)\kappa .} In particular, any constant-curvature space is Einstein and has constant scalar curvature...
different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest...
negative spectrum), and R is the scalarcurvature. This operator often makes an appearance when studying how the scalarcurvature behaves under a conformal change...
and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a...
geometry, which was resolved in the 1980s. It is a statement about the scalarcurvature of Riemannian manifolds: Let (M,g) be a closed smooth Riemannian manifold...
L ∝ R {\displaystyle L\,\propto \,R} where R is the scalarcurvature, a measure of the curvature of space. Almost every theory described in this article...
is a maximally symmetric Lorentzian manifold with constant positive scalarcurvature. It is the Lorentzian[further explanation needed] analogue of an n-sphere...
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