Curvature of Riemannian manifolds

Category:Riemannian geometry 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. 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. The articles Cartan connection and covariant derivative explain two different ways to introduce and calculate the curvature tensor. Curvature of Pseudo-Riemannian manifold can be expressed on the same way with only slight modifications.

Table of contents

1 Ways to express the curvature of a Riemannian manifold 1.1 The curvature tensor 1.1.1 Symmetries and identities 1.2 Sectional curvature 1.3 Curvature form 1.4 The curvature operator 2 Calculation of curvature 3 Further curvature tensors 3.5 Scalar curvature 3.6 Ricci curvature 3.7 Weyl curvature tensor and decomposition of the curvature tensor 4 Related articles

Ways to express the curvature of a Riemannian manifold

The curvature tensor

The curvature of Riemannian manifold can be described by various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connection(or covariant differentiation) and Lie bracket by the following formula:

Here is a linear transformation of the tangent space of the manifold; it is linear in each argument.\nIf and are coordinate vector fields then and therefore the formula simplifies to

i.e. the curvature tensor measures anticommutativity of the covariant derivative. The linear transormation is also called the curvature transformation. \nNB. There are few books where curvature tensor defined with opposite sign.

Symmetries and identities

The curvature tensor has the following symmetries:

The last identity was discovered by Ricci, but is often called the first Bianchi identity, just because it looks similarly to the Bianchi identity below. \nThese three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfis the identities above, one could find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has independent components.\nYet another useful identity follows from these three:

The Bianchi identity (often the second Bianchi identity)\ninvolves the covariant derivatives:

Sectional curvature

Sectional curvature is a further, equivalent but more geometrical, description of the curvature of Riemannian manifolds. It is a function which depends on a section (i.e. a 2-plane in the tangent spaces). It is the Gauss curvature of the -section at p; here -section is a locally-defined piece of surface which has the plane as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of under the exponential map at p. If are two linearly independent vectors in then

The following complex formula indicates that sectional curvature describes the curvature tensor completely:

\n:\n:

Curvature form

The Cartan formalism gives a very elegant way to describe curvature. It is used more for general vector bundles, and for principal bundles, but it works just as well for the tangent bundle with the Levi-Civita connection. The curvature of n-dimensional Riemannian manifold is given by an antisymmetric n×n matrix of 2-forms (or equivalently a 2-form with values in , the Lie algebra of the orthogonal group , which is the structure group of the tangent bundle of a Riemannian manifold). Let be a local section of orthonormal basises. Then one can define the connection form, an antisimmetric matrix of 1-forms which satisfy from the follwowing identity

Then the curvature form is defined by

The following describes relation between curvature form and curvature tensor:

This approach builds in all symmetries of curvature tensor exapt the first Bianchi identity, which takes form

where is an n-vector of 1-forms defined by .\nThe second Bianchi identity takes form

D denotes the exterior covariant derivative

The curvature operator

It is sometimes convenient to think about curvature as an operator \non tangent bivectorss (elements of ), which is uniquely defined by the following identity:\n:

Calculation of curvature

For calculation of curvature

• of hypersurfaces and submanifolds see second fundamental form,\n*in coordinates see covariant derivative,\n*by moving frames see Cartan connection and curvature form.

Further curvature tensors

In general the following tensors and functions do not describe the curvature tensor completely, \nhowever they play important role.

Scalar curvature

Scalar curvature is a function on any Riemannian manifold, usually denoted by Sc. \nIt is the full trace of the curvature tensor; given an orthonormal basis \n in the tangent space at p we have

\nwhere Ric denotes Ricci tensor. The result does not depend on the choice of orthonormal basis.\nStarting with dimension 3, scalar curvature does not describe the curvature tensor completely.

Ricci curvature

Ricci curvature is a linear operator on tangent space at a point, usually denoted by Ric. \nGiven an orthonormal basis \n in the tangent space at p we have

The result does not depend on the choice of orthonormal basis. \nStarting with dimension 4, Ricci curvature does not describe the curvature tensor completely.

Weyl curvature tensor and decomposition of the curvature tensor

The Weyl curvature tensor has the same symmetries as the curvature tensor, plus one extra: its Ricci curvature must vanish.\nThe curvature tensor can be decomposed into the part which depends on the Ricci curvature, and the Weyl tensor. \nIf the dimension n > 3 then the second part can be non-zero. If g′=fg for some positive scalar function f — a conformal change of metric — then W ′ = W. \nFor constant curvature, the Weyl tensor is zero. \nMoreover, W=0 if and only if the metric is locally 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).

Related articles

• Curvature for a more elementary discussion.