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In the
mathematical Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
field of
differential geometry Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in th ...
, the Riemann curvature tensor or Riemann–Christoffel tensor (after
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigo ...
and
Elwin Bruno Christoffel Elwin Bruno Christoffel (; November 10, 1829 – March 15, 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provi ...
) is the most common way used to express the
curvature of Riemannian manifoldsFrom left to right: a surface of negative cylinder),_and_a_surface_of_positive_Gaussian_curvature_(sphere.html" style="text-decoration: none;"class="mw-redirect" title="hyperboloid">Gaussian curvature (hyperboloid), a surface of zero Gaussian cur ...
. It assigns a
tensor In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. Te ...
to each point of a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T'p'M'' at each point ''p''. A common convention is to ta ...
(i.e., it is a
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis o ...
), that measures the extent to which the
metric tensor In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a ...
is not locally isometric to that of Euclidean space. The curvature tensor can also be defined for any
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requ ...
, or indeed any manifold equipped with an
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a ...
. It is a central mathematical tool in the theory of
general relativity General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes spec ...
, the modern theory of
gravity Gravity (), or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or ''gravitate'' toward) one another. On Earth, gravity gives weight to p ...
, and the curvature of
spacetime In physics, spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional manifold. The fabric of space-time is a conceptual model combining the three dimensions of space ...
is in principle observable via the
geodesic deviation equationIn general relativity, geodesic deviation describes the tendency of objects to approach or recede from one another while moving under the influence of a spatially varying gravitational field. Put another way, if two objects are set in motion along tw ...
. The curvature tensor represents the
tidal force The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomena, ...
experienced by a rigid body moving along a
geodesic In geometry, a geodesic () is commonly a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
in a sense made precise by the Jacobi equation. This curvature tensor ''R'' is given in terms of the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique connection on the tangent bundle of a manifold (i.e. affine connection) that preserves the (pseudo- ...
$\nabla$ by the following formula: :$R\left(u, v\right)w = \nabla_u\nabla_v w - \nabla_v \nabla_u w - \nabla_ w$ or equivalently : where 'u'', ''v''is the Lie bracket of vector fields and is a commutator of differential operators. 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. Occasionally, the curvature tensor is defined with the opposite sign. If $u = \partial/\partial x^i$ and $v = \partial/\partial x^j$ are coordinate vector fields then $\left[u, v\right] = 0$ and therefore the formula simplifies to :$R\left(u, v\right)w = \nabla_u\nabla_v w - \nabla_v\nabla_u w .$ The curvature tensor measures ''noncommutativity of the covariant derivative'', and as such is the integrability condition, integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, ''flat'' space). The linear transformation $w \mapsto R\left(u, v\right)w$ is also called the curvature transformation or endomorphism. The curvature formula can also be expressed in terms of the second covariant derivative defined as: : $\nabla^2_ w = \nabla_u\nabla_v w - \nabla_ w$ which is linear in ''u'' and ''v''. Then: : $R\left(u, v\right) = \nabla^2_ - \nabla^2_$ Thus in the general case of non-coordinate vectors ''u'' and ''v'', the curvature tensor measures the noncommutativity of the second covariant derivative.

# Geometric meaning

## Informally

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. This is because tennis courts are built so the surface is flat. On the other hand, the surface of the Earth is curved: we can complete a loop on the surface of the Earth. 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. For this path, first walk to the north pole, then turn 90 degrees and walk down to the equator, and finally turn 90 degrees and walk back to the start. However now the tennis racket will be pointing backwards (towards the east). This process is akin to parallel transporting a vector along the path and the difference identifies how lines which appear "straight" are only "straight" locally. 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. These are the
geodesic In geometry, a geodesic () is commonly a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
of the space, for example any segment of a great circle of a sphere. The concept of a curved space in mathematics differs from conversational usage. For example, if the above process was completed on a cylinder one would find that it is not curved overall as the curvature around the cylinder cancels with the flatness along the cylinder, this is a consequence of Gaussian curvature and the Gauss–Bonnet theorem. A familiar example of this is a floppy pizza slice which will remain rigid along its length if it is curved along its width. The Riemann curvature tensor is a way to capture a measure of the intrinsic curvature. 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).

## Formally

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. However, this property does not hold in the general case. The Riemann curvature tensor directly measures the failure of this in a general
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T'p'M'' at each point ''p''. A common convention is to ta ...
. This failure is known as the non-holonomy of the manifold. Let ''x''''t'' be a curve in a Riemannian manifold ''M''. Denote by τ''x''''t'' : Tx0''M'' → Txt''M'' the parallel transport map along ''x''t. The parallel transport maps are related to the covariant derivative by :$\nabla_ Y = \lim_ \frac\left\left(Y_ - \tau^_\left\left(Y_\right\right)\right\right) = \left.\frac\left\left(\tau_Y\right\right)\_$ for each vector field ''Y'' defined along the curve. Suppose that ''X'' and ''Y'' are a pair of commuting vector fields. Each of these fields generates a one-parameter group of diffeomorphisms in a neighborhood of ''x''0. Denote by τtX and τtY, respectively, the parallel transports along the flows of ''X'' and ''Y'' for time ''t''. Parallel transport of a vector ''Z'' ∈ Tx0''M'' around the quadrilateral with sides ''tY'', ''sX'', −''tY'', −''sX'' is given by :$\tau_^\tau_^\tau_\tau_Z.$ This measures the failure of parallel transport to return ''Z'' to its original position in the tangent space Tx0''M''. Shrinking the loop by sending ''s'', ''t'' → 0 gives the infinitesimal description of this deviation: :$\left.\frac\frac\tau_^\tau_^\tau_\tau_Z\_ = \left\left(\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_\right\right)Z = R\left(X, Y\right)Z$ where ''R'' is the Riemann curvature tensor.

# Coordinate expression

Converting to the tensor index notation, the Riemann curvature tensor is given by :$R^\rho_ = dx^\rho\left\left(R\left\left(\partial_\mu, \partial_\nu\right\right)\partial_\sigma\right\right)$ where $\partial_\mu = \partial/\partial x^\mu$ are the coordinate vector fields. The above expression can be written using Christoffel symbols: :$R^\rho_ = \partial_\mu\Gamma^\rho_ - \partial_\nu\Gamma^\rho_ + \Gamma^\rho_\Gamma^\lambda_ - \Gamma^\rho_\Gamma^\lambda_$ (see also the list of formulas in Riemannian geometry). The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector $A_$ with itself: :$A_ - A_ = -A_ R^_,$ since the Connection (mathematics), connection $\Gamma^\alpha_$ is torsionless, which means that the torsion tensor $\Gamma^\lambda_ - \Gamma^\lambda_$ vanishes. This formula is often called the ''Ricci identity''. This is the classical method used by Gregorio Ricci-Curbastro, Ricci and Tullio Levi-Civita, Levi-Civita to obtain an expression for the Riemann curvature tensor. In this way, the tensor character of the set of quantities $R^_$ is proved. This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows :$\begin &\nabla_\delta \nabla_\gamma T^_ - \nabla_\gamma \nabla_\delta T^_ \\\left[3pt\right] = &R^_ T^_ + \ldots + R^_ T^_ - R^\sigma_ T^_ - \ldots - R^\sigma_ T^_ \end$ This formula also applies to tensor density, tensor densities without alteration, because for the Levi-Civita (''not generic'') connection one gets: :$\nabla_\left\left(\sqrt\right\right) \equiv \left\left(\sqrt\right\right)_ = 0,$ where :$g = \left, \det\left\left(g_\right\right)\.$ It is sometimes convenient to also define the purely covariant version by :$R_ = g_ R^\zeta_.$

# Symmetries and identities

The Riemann curvature tensor has the following symmetries and identities: where the bracket $\langle,\rangle$ refers to the inner product on the tangent space induced by the
metric tensor In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a ...
. The first (algebraic) Bianchi identity was discovered by Gregorio Ricci-Curbastro, Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the Luigi Bianchi, Bianchi identity below. (Also, if there is nonzero Torsion tensor, torsion, the first Bianchi identity becomes a differential identity of the torsion tensor.) . It is often written: $R_ = 0,$where the brackets denote the antisymmetric tensor, antisymmetric part 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. The first 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 $n^2\left\left(n^2 - 1\right\right)/12$ independent components. Interchange symmetry follows from these. The algebraic symmetries are also equivalent to saying that ''R'' belongs to the image of the Young symmetrizer corresponding to the partition 2+2. On a Riemannian manifold one has the covariant derivative $\nabla_u R$ and the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) takes the form of the last identity in the table.

# Ricci curvature

The Ricci curvature tensor is the Tensor_contraction, contraction of the first and third indices of the Riemann tensor. :$\underbrace_ \equiv \underbrace_ = g^ \underbrace_$

# Special cases

## Surfaces

For a two-dimensional surface (topology), 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: :$R_ = f\left(R\right) \left\left(g_g_ - g_g_\right\right)$ and by contracting with the metric twice we find the explicit form: :$R_ = K\left\left(g_g_ - g_g_\right\right) ,$ where $g_$ is the
metric tensor In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a ...
and $K = R/2$ is a function called the Gaussian curvature 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 of the surface. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by :$R_ = Kg_.$

## Space forms

A Riemannian manifold is a space form if its sectional curvature is equal to a constant ''K''. The Riemann tensor of a space form is given by :$R_ = K\left\left(g_g_ - g_g_\right\right).$ 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.