Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann

Bernhard Riemann

was an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...

 and Elwin Bruno Christoffel

Elwin Bruno Christoffel

Elwin Bruno Christoffel was a German mathematician and physicist.-Life:...

, is the most standard way to express curvature of Riemannian manifolds

Curvature of Riemannian 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. Riemann introduced an abstract and rigorous way to define it, now known as the curvature tensor...

. It associates a tensor

Tensor

Tensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...

 to each point of a Riemannian manifold

Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor...

 (i.e., it is a tensor field

Tensor field

In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...

), that measures the extent to which the metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 is not locally isometric to a Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...

, or indeed any manifold equipped with an affine connection

Affine connection

In the mathematical field of differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...

.

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann

Bernhard Riemann

was an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...

 and Elwin Bruno Christoffel

Elwin Bruno Christoffel

Elwin Bruno Christoffel was a German mathematician and physicist.-Life:...

, is the most standard way to express curvature of Riemannian manifolds

Curvature of Riemannian 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. Riemann introduced an abstract and rigorous way to define it, now known as the curvature tensor...

. It associates a tensor

Tensor

Tensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...

 to each point of a Riemannian manifold

Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor...

 (i.e., it is a tensor field

Tensor field

In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...

), that measures the extent to which the metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 is not locally isometric to a Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...

, or indeed any manifold equipped with an affine connection

Affine connection

In the mathematical field of differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...

. It is a central mathematical tool in the theory of general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...

, the modern theory of gravity, and the curvature of spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...

 is in principle observable via the geodesic deviation equation

Geodesic deviation equation

In general relativity, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring geodesics or, equivalently, the tidal force experienced by a rigid body moving along a geodesic...

. The curvature tensor represents the tidal force

Tidal force

The tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational force exerted on one body by a second body is not constant across its diameter...

 experienced by a rigid body moving along a geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".In the presence of a metric, geodesics are defined to be the shortest path between points on the space...

 in a sense made precise by the Jacobi equation

Jacobi field

In Riemannian geometry, a Jacobi field is a vector field along a geodesic in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all...

.

The curvature tensor is given in terms of the Levi-Civita connection

Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric....

  by the following formula:
where [u,v] is the Lie bracket of vector fields

Lie bracket of vector fields

In the mathematical field of differential topology, the Lie bracket of vector fields,Jacobi–Lie bracket, or Commutator of vector fields is a bilinear differential operator which assigns, to any two vector fields X and Y on a smooth manifold M, a third vector field denoted [X, Y]...

. For each pair of tangent vectors u, v, R(u,v) is a linear transformation of the tangent space of the manifold. It is linear in u and v, and so defines a tensor. Occassionally, the curvature tensor is defined with the opposite sign. If and are coordinate vector fields then and therefore the formula simplifies to
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). The linear transformation is also called the curvature transformation or endomorphism.

Geometrical meaning

In a Euclidean space, when a vector is parallel transport

Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...

ed around a loop, it will always return to its original position. The Riemann curvature tensor directly measures the failure of this in a general Riemannian manifold

Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor...

; this failure is known as the holonomy

Holonomy

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...

 of the manifold.

Let x_t be a curve in a Riemannian manifold M. Denote by τ_t : T_x0M → T_xtM the parallel transport map along x_t. The parallel transport maps are related to the covariant derivative

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach...

 by
for each vector field

Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of...

 Y defined along the curve.

Suppose that X and Y are a pair of commuting vector fields. Each of these fields generates a pair of one-parameter groups of diffeomorphisms in a neighborhood of x₀. Denote by τ_tX and τ_tY, respectively, the parallel transports along the flows of X and Y for time t. Parallel transport of a vector Z ∈ T_x0M around the quadrilateral with sides tY, sX, −tY, −sX is given by
This measures the failure of parallel transport to return Z to its original position in the tangent space T_x0M. Shrinking the loop by sending s, t → 0 gives the infinitesimal description of this deviation:
where R is the Riemann curvature tensor.

Coordinate expression

In local coordinates

Local coordinates

Local coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.Local...

  the Riemann curvature tensor is given by
where are the coordinate vector fields. The above expression can be written using Christoffel symbols

Christoffel symbols

In mathematics and physics, the Christoffel symbol describes curvature in a non-euclidean space, such as geometry on the surface of the globe. Whereas the metric tensor describes a sort of first derivative of the warp, the Christoffel symbol describes a sort of second derivative...

:

(see also the list of formulas in Riemannian geometry).

Also define the purely covariant version by

Symmetries and identities

The Riemann curvature tensor has the following symmetries:

The last identity was discovered by Ricci

Gregorio Ricci-Curbastro

Gregorio Ricci-Curbastro was an Italian mathematician. He was born at Lugo di Romagna. He is most famous as the inventor of the tensor calculus but published important work in many fields....

, but is often called the first Bianchi identity or algebraic 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 independent components.

Yet another useful identity follows from these three:

The Bianchi identity (often called the second Bianchi identity or differential Bianchi identity)
involves the covariant derivative:
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:

Skew symmetry

Interchange symmetry

First Bianchi identity

This is often written

where the brackets denote the antisymmetric part

Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it flips sign when the two indices are interchanged:An antisymmetric tensor is a tensor for which there are two indices on which it is antisymmetric...

 on the indicated indices. This is equivalent to the previous version of the identity because the Riemann tensor is already skew on its last two indices.

Second Bianchi identity

The semi-colon denotes a covariant derivative. Equivalently,

again using the antisymmetry on the last two indices of R.

Special cases

Surfaces
For a two-dimensional surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R³ — for example, the surface of a ball...

, the Bianchi identities imply that the Riemann tensor can be expressed as
where is the metric tensor

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 and is a function called the Gaussian curvature

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ₁ and κ₂, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the...

 and a, b, c and d take values either 1 or 2. The Riemann tensor has only one functionally independent component. The Gaussian curvature coincides with the sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σ_p in the tangent space at p...

 of the surface. It is also exactly half the scalar curvature

Scalar curvature

In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

 of the 2-manifold, while the Ricci curvature

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...

 tensor of the surface is simply given by
Space forms
A Riemannian manifold is a space form

Space form

In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K.-Reduction to generalized crystallography:...

 if its sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σ_p in the tangent space at p...

 is equal to a constant K. The Riemann tensor of a space form is given by
Conversely, except in dimension 2, if the curvature of a Riemannian manifold has this form for some function K, then the Bianchi identities imply that K is constant and thus that the manifold is (locally) a space form.

See also

• Basic introduction to the mathematics of curved spacetime
  Basic introduction to the mathematics of curved spacetime
  An understanding of calculus and differential equations is necessary for the understanding of nonrelativistic physics. In order to understand special relativity one also needs an understanding of tensor calculus...
  
  
• Decomposition of the Riemann curvature tensor