Curvature of Riemannian manifolds
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 of surfaces and other objects.
The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
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)
i.e. the curvature tensor measures noncommutativity of the covariant derivative.
There are a few books where the curvature tensor is defined with opposite sign.
The first two should be addressed as antisymmetry and Lie algebra property respectively, since the second means that the
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 dilations.
It has strong ties with the theory of Lie groups and algebras, Lie triples and Jordan algebras.
The three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor that satisfies the identities above, one could find a Riemannian manifold with such a curvature tensor at some point.
-section is a locally defined piece of surface that has the plane
The connection form gives an alternative way to describe curvature.
, which is the structure group of the tangent bundle of a Riemannian manifold).
Then one can define the connection form, an antisymmetric matrix of 1-forms
This approach builds in all symmetries of curvature tensor except the first Bianchi identity, which takes form
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).
In general the following tensors and functions do not describe the curvature tensor completely; however they play an important role.
Scalar curvature is a function on any Riemannian manifold, denoted variously by
It is the full trace of the curvature tensor; given an orthonormal basis
The result does not depend on the choice of orthonormal basis.
Ricci curvature is a linear operator on tangent space at a point, usually denoted by
The result does not depend on the choice of orthonormal basis.
Explicit expressions for the Ricci tensor in terms of the Levi-Civita connection is given in the article on Christoffel symbols.
For a manifold of constant curvature, the Weyl tensor is zero.
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
