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Concept
Curvature of Riemannian manifolds
Summary
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.
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) 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.
If and are coordinate vector fields then and therefore the formula simplifies to
i.e. the curvature tensor measures noncommutativity of the covariant derivative.
The linear transformation is also called the curvature transformation or endomorphism.
NB. There are a few books where the curvature tensor is defined with opposite sign.
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).
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