In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. The curvature tensor is given in terms of a Levi-Civita connection (more generally, an affine connection) (or covariant differentiation) by the following formula: In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... 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. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... // Mathmatics In mathematics, the term torsion has several meanings, mostly unrelated to each other. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...

Here R(u,v) is a linear transformation of the tangent space of the manifold; it is linear in each argument.

NB. Some authors define the curvature tensor with the opposite sign.

If and are coordinate vector fields then [u,v] = 0 and therefore the formula simplifies to

i.e. the curvature tensor measures anticommutativity of the covariant derivative.

The linear transformation is also called the curvature transformation.

Symmetries and identities

The Riemann curvature tensor has the following symmetries:

The last identity was discovered by Ricci, but is often called the first Bianchi identity, because it looks similar to the Bianchi identity below. These three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has n²(n² − 1) / 12 independent components. Yet another useful identity follows from these three: Gregorio Ricci-Curbastro (January 12, 1853 - August 6, 1925) was an Italian mathematician. ...

The Bianchi identity (often the second Bianchi identity) involves the covariant derivatives:

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as: In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

where the square brackets denote symmetrisation over the indices and the semi-colon is a covariant derivative. These identities find application in physics, especially general relativity. The third and fourth identities are sometimes called the algebraic Bianchi identity and the differential Bianchi identity, respectively. Two-dimensional visualization of space-time distortion. ...

See also

• Curvature of Riemannian manifolds
• Sectional curvature
• Curvature form