NationMaster - Encyclopedia: Scalar curvature

In Riemannian geometry, the scalar curvature (or 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. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, curvature refers to a number of loosely related concepts in different areas of geometry. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

In two dimensions the scalar curvature completely characterizes the curvature of a Riemannian manifold. In dimensions ≥ 3, however, more information is needed. See curvature of Riemannian manifolds for a complete discussion. In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ...

The scalar curvature usually denoted by S (other notation are Sc, R). It is defined as the trace of the Ricci curvature tensor with respect to the metric: In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

$S = mbox{tr}_g,operatorname{Ric}$

The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first raise an index to obtain a (1,1)-valent tensor in order take the trace. In terms of local coordinates one can write Local coordinates are measurement indices into a local coordinate system or a local coordinate space. ...

S = g i j R i j

where

$operatorname{Ric} = R_{ij},dx^iotimes dx^j$

Given a coordinate system and a metric tensor, scalar curvature can be expressed as follows

$S = g^{ab} (Gamma^c_{ab,c} - Gamma^c_{ac,b} + Gamma^c_{ab}Gamma^d_{cd} - Gamma^d_{ac} Gamma^c_{bd})$

where $Gamma^a_{bc}$ are the Christoffel symbols of the metric. In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829â€“1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

Unlike the Riemann curvature tensor or the Ricci tensor, which both can be naturally be defined for any affine connection, the scalar curvature is entirely special to the realm of Riemannian geometry; its very definition involves the metric in an inextricable fashion. In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ...

1 Direct geometric interpretation
2 2 dimensions
3 Traditional notation
4 See also

Direct geometric interpretation

When the scalar curvature is positive at a point, the volume of a small ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is instead larger than it would be in Euclidean space.

This can be made more quantitative, in order to characterize the precise value of the scalar curvature S at a point p of a Riemannian n-manifold $(M, g)$ . Namely, the ratio of the n-dimensional volume of a ball of radius ε in the manifold to that of a corresponding ball in Euclidean space is given, for small ε, by

$frac{operatorname{Vol} (B_varepsilon(p) subset M)}{operatorname{Vol} (B_varepsilon(0)subset {mathbb R}^n)}= 1- frac{S}{6(n+2)}varepsilon^2 + O(varepsilon^4)$

Thus, the second derivative of this ratio, evaluated at radius ε = 0, is exactly minus the scalar curvature divided by 3(n + 2).

Boundaries of these balls are (n-1) dimensional spheres with radii $ε$ ; their areas satisfy the following equation:

$frac{operatorname{Area} (partial B_varepsilon(p) subset M)}{operatorname{Area} (partial B_varepsilon(0)subset {mathbb R}^n)}= 1- frac{S}{6n}varepsilon^2 + O(varepsilon^4)$

2 dimensions

In 2 dimensions, scalar curvature is exactly twice the Gauss curvature: From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere). ...

$S = frac{2}{rho_1rho_2}$

where $rho_1,,rho_2$ are principal radii of the surface. For example, scalar curvature of a sphere with radius r is equal to $2/r^2,$ . More generally, scalar curvature of an n-sphere with a radius r is $n(n-1)/r^2,$ . Principal curvature is the inverse of the radius of the osculating circle. ...

2-dimensional Riemann tensor has only one independent component and it can be easily expressed in terms of the scalar curvature and metric area form. In any coordinate system, one thus has: In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

$2R_{1212} ,= S det (g_{ij}) = S[g_{11}g_{22}-(g_{12})^2]$

Traditional notation

Among those who use index notation for tensors, it is common to use the letter R to represent three different things:

the Riemann curvature tensor: $R_{ijk}^l$ or $R a b c d$
the Ricci tensor: $R i j$
the scalar curvature: R

These three are then distinguished from each other by their number of indices: the Riemann tensor has four indices, the Ricci tensor has two indices, and the Ricci scalar has zero indices. Those not using an index notation usually reserve R for the full Riemann curvature tensor. In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...

Scalar curvature - definition of Scalar curvature in Encyclopedia (172 words)
	In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold.
	In two dimensions the scalar curvature completely characterizes the curvature of a Riemannian manifold.
	The scalar curvature is defined as the trace of the Ricci curvature with respect to the metric:

Curvature of Riemannian manifolds - Wikipedia, the free encyclopedia (941 words)
	It is the Gauss curvature of the σ-section at p; here σ-section is a locally-defined piece of surface which has the plane σ as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of σ under the exponential map at p.
	It is possible to do this precisely because of the symmetries of the curvature tensor (namely antisymmetry in the first and last pairs of indices, and block-symmetry of those pairs).
	For a manifold of constant curvature, the Weyl tensor is zero.

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Direct geometric interpretation var ACE_AR = {Site: '756743', Size: '300250'};

2 dimensions

Traditional notation

See also

Direct geometric interpretation