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Scalar curvature
 
 
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In Riemannian geometry
Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, manifold with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smooth function from point to point....
, the scalar curvature (or Ricci scalar) is the simplest curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 invariant of a Riemannian manifold
Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
. To each point on a Riemannian manifold, it assigns a single real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
 determined by the intrinsic geometry of the manifold near that point.

In two dimensions the scalar curvature completely characterizes the curvature of a Riemannian manifold. In dimensions ≥ 3, however, more information is needed.






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In Riemannian geometry
Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, manifold with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smooth function from point to point....
, the scalar curvature (or Ricci scalar) is the simplest curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 invariant of a Riemannian manifold
Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
. To each point on a Riemannian manifold, it assigns a single real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
 determined by the intrinsic geometry of the manifold near that point.

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
Curvature of Riemannian manifolds

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....
 for a complete discussion.

The scalar curvature is usually denoted by S (other notations are Sc, R). It is defined as the trace
Trace (linear algebra)

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 of A, i.e.,...
 of the Ricci curvature
Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space....
 tensor with respect to the metric
Metric tensor

In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
:

The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first raise an index
Raising and lowering indices

In mathematics and mathematical physics, given a tensor on a manifold M, in the presence of a nonsingular form on M , one can raise or lower indices: change a tensor to a tensor or to a tensor ....
 to obtain a (1,1)-valent tensor in order to take the trace. In terms of local coordinates
Local coordinates

Local coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc....
 one can write

where

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

where are the Christoffel symbols
Christoffel symbols

In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor....
 of the metric.

Unlike the Riemann curvature tensor
Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann?Christoffel tensor is the most standard way to express curvature of Riemannian manifolds....
 or the Ricci tensor, which both can be naturally be defined for any affine connection
Affine connection

In the mathematics of differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits vector field to be derivative as if they were functions on the manifold with values in a fixed vector space....
, the scalar curvature is entirely special to the realm of Riemannian geometry; its very definition involves the metric in an inextricable fashion.

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 . 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



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 ; their areas satisfy the following equation:



2 dimensions


In 2 dimensions, scalar curvature is exactly twice the Gauss curvature:

where are principal radii
Principal curvature

In differential geometry, the two principal curvatures at a given point of a surface measure how the surface bends by different amounts in different directions at that point....
 of the surface. For example, scalar curvature of a sphere with radius r is equal to . More generally, scalar curvature of an n-sphere with a radius r is .

The 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:

Traditional notation


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

    In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann?Christoffel tensor is the most standard way to express curvature of Riemannian manifolds....
    : or
  2. the Ricci tensor:
  3. 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.

See also

  • Basic introduction to the mathematics of curved spacetime
    Basic introduction to the mathematics of curved spacetime

    An understanding of calculus and differential equations is necessary for the understanding of nonrelativistic physics. In order to understand special relativity one also needs an understanding of tensor calculus....