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In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry.
Contents
- The Riemann curvature tensor
- Symmetries and identities
- Sectional curvature
- Curvature form
- The curvature operator
- Further curvature tensors
- Scalar curvature
- Ricci curvature
- Weyl curvature tensor
- Ricci decomposition
- Calculation of curvature
- References
For a more elementary discussion see the article on curvature which discusses the curvature of curves and surfaces in 2 and 3 dimensions, as well as the differential geometry of surfaces.
The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
The Riemann curvature tensor
The curvature of a Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connection (or covariant differentiation)
Here
i.e. the curvature tensor measures noncommutativity of the covariant derivative.
The linear transformation
NB. There are a few books where the curvature tensor is 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 similar to the Bianchi identity below. The first two should be addressed as antisymmetry and Lie algebra property respectively, since the second means, that the R(u, v) for all u, v are elements of the pseudo-orthogonal Lie algebra. All three together should be named pseudo-orthogonal curvature structure. They give rise to a tensor only by identifications with objects of the tensor algebra - but likewise there are identifications with concepts in the Clifford-algebra. Let us note, that these three axioms of a curvature structure give rise to a well-developed structure theory, formulated in terms of projectors (a Weyl projector, giving rise to Weyl curvature and an Einstein projector, needed for the setup of the Einsteinian gravitational equations). This structure theory is compatible with the action of the pseudo-orthogonal groups plus dilatations. It has strong ties with the theory of Lie groups and algebras, Lie triples and Jordan algebras. See the references given in the discussion.
The three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies 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
The Bianchi identity (often the second Bianchi identity) involves 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
If
The following formula indicates that sectional curvature describes the curvature tensor completely:
Or in a simpler formula:
Curvature form
The connection form gives an alternative 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
Let
Then the curvature form
Note that the expression "
This approach builds in all symmetries of curvature tensor except the first Bianchi identity, which takes form
where
D denotes the exterior covariant derivative
The curvature operator
It is sometimes convenient to think about curvature as an operator
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).
Further curvature tensors
In general the following tensors and functions do not describe the curvature tensor completely, however they play an important role.
Scalar curvature
Scalar curvature is a function on any Riemannian manifold, usually denoted by Sc. It is the full trace of the curvature tensor; given an orthonormal basis
where Ric denotes Ricci tensor. The result does not depend on the choice of orthonormal basis. Starting 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. Given an orthonormal basis
The result does not depend on the choice of orthonormal basis. With four or more dimensions, Ricci curvature does not describe the curvature tensor completely.
Explicit expressions for the Ricci tensor in terms of the Levi-Civita connection is given in the article on Christoffel symbols.
Weyl curvature tensor
The Weyl curvature tensor has the same symmetries as the curvature tensor, plus one extra: its trace (as used to define the Ricci curvature) must vanish. In dimensions 2 and 3 Weyl curvature vanishes, but if the dimension n > 3 then the second part can be non-zero.
Ricci decomposition
Although individually, the Weyl tensor and Ricci tensor do not in general determine the full curvature tensor, the Riemann curvature tensor can be decomposed into a Weyl part and a Ricci part. This decomposition is known as the Ricci decomposition, and plays an important role in the conformal geometry of Riemannian manifolds. In particular, it can be used to show that if the metric is rescaled by a conformal factor of
where
Calculation of curvature
For calculation of curvature