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. 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. Ways to express the curvature of a Riemannian manifold. 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) formula_1 and Lie bracket formula_2 by the following formula: Here formula_4 is a linear transformation of the tangent space of the manifold; it is linear in each argument. If formula_5 and formula_6 are coordinate vector fields then formula_7 and therefore the formula simplifies to  i.e. the curvature tensor measures "noncommutativity of the covariant derivative". The linear transformation formula_9 is also called the curvature transformation or endomorphism. 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 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 dilations. 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 formula_13 independent components. Yet another useful identity follows from these three: 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 formula_16 which depends on a "section" formula_17 (i.e. a 2-plane in the tangent spaces). It is the Gauss curvature of the formula_18-"section" at "p"; here formula_18-"section" is a locally defined piece of surface which has the plane formula_18 as a tangent plane at "p", obtained from geodesics which start at "p" in the directions of the image of formula_18 under the exponential map at "p". If formula_22 are two linearly independent vectors in formula_17 then The following formula indicates that sectional curvature describes the curvature tensor completely : Or in a simpler formula: formula_28 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 an "n"-dimensional Riemannian manifold is given by an antisymmetric "n"×"n" matrix formula_29 of 2-forms (or equivalently a 2-form with values in formula_30, the Lie algebra of the orthogonal group formula_31, which is the structure group of the tangent bundle of a Riemannian manifold). Let formula_32 be a local section of  orthonormal bases. Then one can define the connection form, an antisymmetric matrix of 1-forms formula_33 which satisfy from the following identity Then the curvature form formula_35 is defined by Note that the expression "formula_37" is short hand for formula_38 and hence does not necessarily vanish. The following describes relation between curvature form and curvature tensor: This approach builds in all symmetries of curvature tensor except the "first Bianchi identity", which takes form where formula_41 is an "n"-vector of 1-forms defined by formula_42. The "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 formula_44 on tangent bivectors (elements of formula_45), which is uniquely defined by the following identity: 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, denoted variously by formula_47 or formula_48. It is the full  trace of the curvature tensor; given an orthonormal basis formula_49 in the tangent space at a point we have where formula_51 denotes the 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 "formula_51". Given an orthonormal basis formula_49 in the tangent space at "p" we have 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 Riemann curvature tensor, but with one extra constraint: its trace (as used to define the Ricci curvature) must vanish. The Weyl tensor is invariant with respect to a conformal change of metric: if two metrics are related as formula_55 for some positive scalar function formula_56, then formula_57. In dimensions 2 and 3 the Weyl tensor vanishes, but in 4 or more  dimensions the Weyl tensor can be non-zero. For a manifold of constant curvature, the Weyl tensor is zero. Moreover, formula_58 if and only if the metric is locally conformal to the Euclidean metric. 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 formula_59, then the Riemann curvature tensor changes to (seen as a (0, 4)-tensor): where formula_61 denotes the Kulkarni–Nomizu product and Hess is the Hessian. Calculation of curvature. For calculation of curvature