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Curvature tensor

In differential geometry, the curvature tensor is one of the most important notions. Gauss was the first to consider curvature, showing that the product of the principal curvatures of curves is an intrinsic property of a surface. Riemann generalized this notion to n-dimensional Riemannian manifolds. Later, the curvature tensor was understood in terms of connectionss.

Golden triangle of Riemannian geometry

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.

Various forms of curvature tensor on a Riemannian manifold

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.

Scalar and Ricci curvatures

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.

Weyl curvature and decomposition of the curvature tensor

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.

If g'=fg for some scalar function - the conformal change of metric - then W'=fW. For constant curvature, the Weyl tensor is zero. Moreover, W=0 if and only if metric is conformal to the standard Euclidean metric (equal to fg, where g is the standard metric in some coordinate frame and f is some scalar function). The curvature is constant if and only if W=0 and Ric=s/n