Scalar curvature

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).

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.

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.

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.

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.

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

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.

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.

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.

The Yamabe problem was solved by Trudinger, Aubin, and Schoen.

can be conformally rescaled to produce a manifold of constant scalar curvature.

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.

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

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

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.

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.

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.

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

# The scalar curvature:

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

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

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

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.

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.

In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface.

In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.