Curvature tensor
In differential geometry, the curvature tensor is one of the most important notions; it generalizes Gauss curvature to higher dimensions. The infinitesimal geometry of Riemannian manifolds with dimension ≥ 3 is too complicated to be described by one number at a given point. Riemann introduced a way to describe it as a "little monster tensor". Similar notions have found applications everywhere in differential geometry.What follows is a description of this tensor; the reader is assumed to be familiar with Gauss curvature. Articles Cartan connection and Covariant derivative contain two different ways to introduce and calculate the curvature tensor.
Connectionss, parallel transport and curvature form the so-called golden triangle of Riemannian geometry. All three of these give equivalent structure - parallel transport is equivalent to specifying a covariant way of differentiating - or a connection; and a connection determines the curvature tensor.
The curvature tensor is given in terms of a connection (covariant differentiation) by the following formula:
The curvature tensor, on the other hand, via holonomy, determines parallel transport, although only up to a gauge.
There are several equivalent ways to think of the curvature tensor in the case of a Riemannian manifold. Perhaps the easiest way to understand it is as a linear transformation R of 2-forms.
To explain this, consider the sectional curvature, i.e. the curvature of a two-dimensional geodesic surface passing through a point - a section, which is the image of some tangent plane under the exponential map. The corresponding tangent plane can be represented by a 2-form. The curvature tensor gives information equivalent to specifying all sectional curvatures. The squared norm of a 2-form times the corresponding sectional curvature in fact gives a new quadratic form on a space of 2-forms, and it is given precisely by the symmetric linear operator R. In other words, (R(s),s)=k(s)(s,s).
The operator R can be understood in another way. Each 2-form s can be represented by a small rectangular loop (in many ways, but the corresponding form is what matters here). Then parallel transport around this loop gives rise to a transformation of the tangent space. This is a infinitesimal transformation of the tangent space, which can be represented by an element of the lie algebra corresponding to the lie group of all linear transformations of the tangent space. But this lie algebra is again an algebra of 2-forms, and R(s) is just this generator.
The lie algebra of all the loop transformations is the lie algebra of theholonomy corresponding to curvature.
Another way to represent curvature is as a (1,3)-valent tensor. In Riemannian geometry, valence of this tensor can be altered, and there are other equivalent representations of curvature.
In the Cartan formalism, the curvature is given as a matrix of 2-forms.
In two dimensions, the curvature tensor is determined by the scalar curvature - which is the full trace of the curvature tensor.
In three dimensions, the curvature tensor is specified by the Ricci curvature - which is a partial trace of the curvature tensor. This has to do with the fact that space of 2-forms is three-dimensional: the same reason why we can define the vector product for 3 dimensions (the vector product is just the wedge product of two 1-forms composed with the Hodge star operator, if we represent vectors with their corresponding 1-forms).
In higher dimensions, the full curvature tensor contains more information than
the Ricci curvature.
For dimension n>3, the curvature tensor can be decomposed into the part which depends on the Ricci curvature, and the Weyl tensor. If R is the (0,4) valent tensor version of the Riemannian curvature, then
where Ric is the (0,2) valent version of the Ricci curvature, s is
scalar curvature and g is (0,2) valent metric tensor and
is the so called Kulkarni-Nomizu product of the two (0,2) tensors.
Golden triangle of Riemannian geometry
Various forms of curvature tensor on a Riemannian manifold
Scalar and Ricci curvatures
Weyl curvature and decomposition of the curvature tensor