Scalar curvature

Ricci scalarcurvaturecurvature scalarRicci scalar curvatureRicci curvature scalarbelownegatively curvedRicci curvature scalar RscalarTrace of a tensor with respect to a metric tensor
In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold.wikipedia
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Einstein–Hilbert action

Einstein-Hilbert actionEinstein–Hilbert LagrangianEinstein-Hilbert Lagrangian
In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action.
where is the determinant of the metric tensor matrix, R is the Ricci scalar, and is Einstein's constant (G is the gravitational constant and c is the speed of light in vacuum).

Positive energy theorem

positive mass conjecturepositive mass theorem
One such result is the positive mass theorem of Schoen, Yau and Witten.
The theorem is a scalar curvature comparison theorem, with asymptotic boundary conditions, and a corresponding statement of geometric rigidity.

Einstein field equations

Einstein field equationEinstein's field equationsEinstein's field equation
The Euler–Lagrange equations for this Lagrangian under variations in the metric constitute the vacuum Einstein field equations, and the stationary metrics are known as Einstein metrics.
where R is the Ricci curvature tensor, R is the scalar curvature, g is the metric tensor, Λ is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and T is the stress–energy tensor.

Shing-Tung Yau

YauS.-T. YauYau, Shing-Tung
One such result is the positive mass theorem of Schoen, Yau and Witten.
This has motivated the work of Simon Donaldson on scalar curvature and stability.

Lagrangian (field theory)

LagrangianLagrangian densityLagrangian field theory
In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action.
:R is the curvature scalar, which is the Ricci tensor contracted with the metric tensor, and the Ricci tensor is the Riemann tensor contracted with a Kronecker delta.

Einstein manifold

Einstein metricEinsteinEinstein manifolds
The Euler–Lagrange equations for this Lagrangian under variations in the metric constitute the vacuum Einstein field equations, and the stationary metrics are known as Einstein metrics.
Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by

Richard Schoen

Richard M. SchoenRichard Melvin SchoenRick Schoen
One such result is the positive mass theorem of Schoen, Yau and Witten.
positive function) so as to produce a metric of constant scalar curvature.

Curvature

curvednegative curvatureextrinsic curvature
In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold.
Two more generalizations of curvature are the scalar curvature and Ricci curvature.

General relativity

general theory of relativitygeneral relativity theoryrelativity
In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action.
In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the Ricci scalar, take on infinite values.

Thierry Aubin

AubinAubin, Thierry
The Yamabe problem was solved by Trudinger, Aubin, and Schoen.
can be conformally rescaled to produce a manifold of constant scalar curvature.

Ricci curvature

Ricci tensorRicci curvature tensorTrace-free Ricci tensor
The scalar curvature of an n-manifold is defined as the trace of the Ricci tensor, and it can be defined as n(n − 1) times the average of the sectional curvatures at a point. The scalar curvature S (commonly also R, or Sc) is defined as the trace of the Ricci curvature tensor with respect to the metric:
is the Ricci tensor, S is the scalar curvature, g is the metric tensor, and n is the dimension of M.

Trace (linear algebra)

tracetracelessmatrix trace
The scalar curvature S (commonly also R, or Sc) is defined as the trace of the Ricci curvature tensor with respect to the metric:

Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
Unlike the Riemann curvature tensor or the Ricci tensor, both of which can be defined for any affine connection, the scalar curvature requires a metric of some kind.
It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by

Prescribed scalar curvature problem

In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold M and a smooth, real-valued function ƒ on M, construct a Riemannian metric on M whose scalar curvature equals ƒ.

Yamabe invariant

where R_g is the scalar curvature of g and dV_g is the volume density associated to the metric g. The exponent in the denominator is chosen so that the functional is scale-invariant: for every positive real constant c, it satisfies.

Vermeil's theorem

In differential geometry, Vermeil's theorem essentially states that the scalar curvature is the only (non-trivial) absolute invariant among those of prescribed type suitable for Einstein’s theory.

Kretschmann scalar

Kretschmann invariant
where R^{ab} is the Ricci curvature tensor and R is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor).

Introduction to the mathematics of general relativity

Basic introduction to the mathematics of curved spacetimenon-Euclidean geometry of curved space-time
# The scalar curvature:

Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold.

Real number

realrealsreal-valued
To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point.

Volume

volumetriccapacityOrders of magnitude (volume)
Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space.

Euclidean space

EuclideanspaceEuclidean vector space
Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space.

Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem
In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface.

Curvature of Riemannian manifolds

curvatureabstract definition of curvaturecurvature of a Riemannian manifold
In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.