Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds.

It is a local invariant of Riemannian metrics that measures the failure of the second covariant derivatives to commute.

A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space.

The curvature of spacetime is in principle observable via the geodesic deviation equation.

The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.

It turns out that the right-hand side actually only depends on the value of the vector fields

The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space).

Since the Levi-Civita connection is torsion-free, its curvature can also be expressed in terms of the second covariant derivative[3] which depends only on the values of

The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector

[6] This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor.

[7] This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows [8] This formula also applies to tensor densities without alteration, because for the Levi-Civita (not generic) connection one gets:[6] where It is sometimes convenient to also define the purely covariant version of the curvature tensor by One can see the effects of curved space by comparing a tennis court and the Earth.

Start at the lower right corner of the tennis court, with a racket held out towards north.

Then while walking around the outline of the court, at each step make sure the tennis racket is maintained in the same orientation, parallel to its previous positions.

Once the loop is complete the tennis racket will be parallel to its initial starting position.

Starting at the equator, point a tennis racket north along the surface of the Earth.

Once again the tennis racket should always remain parallel to its previous position, using the local plane of the horizon as a reference.

Each time a loop is completed the tennis racket will be deflected further from its initial position by an amount depending on the distance and the curvature of the surface.

It is possible to identify paths along a curved surface where parallel transport works as it does on flat space.

The concept of a curved space in mathematics differs from conversational usage.

A familiar example of this is a floppy pizza slice, which will remain rigid along its length if it is curved along its width.

When you write it down in terms of its components (like writing down the components of a vector), it consists of a multi-dimensional array of sums and products of partial derivatives (some of those partial derivatives can be thought of as akin to capturing the curvature imposed upon someone walking in straight lines on a curved surface).

When a vector in a Euclidean space is parallel transported around a loop, it will again point in the initial direction after returning to its original position.

The Riemann curvature tensor directly measures the failure of this in a general Riemannian manifold.

The parallel transport maps are related to the covariant derivative by for each vector field

Each of these fields generates a one-parameter group of diffeomorphisms in a neighborhood of

The above expression can be written using Christoffel symbols: (See also List of formulas in Riemannian geometry).

refers to the inner product on the tangent space induced by the metric tensor and the brackets and parentheses on the indices denote the antisymmetrization and symmetrization operators, respectively.

The algebraic symmetries are also equivalent to saying that R belongs to the image of the Young symmetrizer corresponding to the partition 2+2.

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor has only one independent component, which means that the Ricci scalar completely determines the Riemann tensor.

There is only one valid expression for the Riemann tensor which fits the required symmetries: and by contracting with the metric twice we find the explicit form: where

Figure showing the geometric meaning of the Riemann curvature tensor in a spherical curved manifold. The fact that this transfer can define two different arrows at the starting point gives rise to the Riemann curvature tensor. The orthogonal symbol indicates that the dot product (provided by the metric tensor) between the transmitted arrows (or the tangent arrows on the curve) is zero. The angle between the two arrows is zero when the space is flat and greater than zero when the space is curved. The more curved the space, the greater the angle.