In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, 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 ...
and
Elwin Bruno Christoffel
Elwin Bruno Christoffel (; 10 November 1829 – 15 March 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 manifolds. 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. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
to each point of a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
(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 ...
). It is a local invariant of Riemannian metrics which 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. 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 ...
, or indeed any manifold equipped with an
affine connection.
It is a central mathematical tool in the theory of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the modern theory of
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, and the curvature of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
is in principle observable via the
geodesic deviation equation. 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 phenomen ...
experienced by a rigid body moving along a
geodesic
In geometry, a geodesic () is 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 connecti ...
in a sense made precise by the
Jacobi equation.
Definition
Let (''M'', g) be a
Riemannian or
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 ...
, and
be the space of all
vector fields on M. We define the ''Riemann curvature tensor'' as a map
by the following formula where
is an
affine connection:
:
or equivalently
:
where
'X'', ''Y''is the
Lie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields ''X'' and ''Y'' on a smooth ...
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
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.
The curvature formula can also be expressed in terms of the
second covariant derivative
In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields ...
defined as:
:
which is linear in ''u'' and ''v''. Then:
:
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 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 connecti ...
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
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
...
and Gauss'
Theorema Egregium. 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 , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
. This failure is known as the non-
holonomy of the manifold.
Let
be a curve in a Riemannian manifold
. Denote by
the parallel transport map along
. The parallel transport maps are related to the
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 differe ...
by
:
for each
vector field defined along the curve.
Suppose that
and
are a pair of commuting vector fields. Each of these fields generates a one-parameter group of diffeomorphisms in a neighborhood of
. Denote by
and
, respectively, the parallel transports along the flows of
and
for time
. Parallel transport of a vector
around the quadrilateral with sides
,
,
,
is given by
:
This measures the failure of parallel transport to return
to its original position in the tangent space
. Shrinking the loop by sending
gives the infinitesimal description of this deviation:
:
where
is the Riemann curvature tensor.
Coordinate expression
Converting to the
tensor index notation, the Riemann curvature tensor is given by
:
where
are the coordinate vector fields. The above expression can be written using
Christoffel symbols:
:
(see also the
list of formulas in Riemannian geometry
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwhise.
Christoffel symbols, covariant deriva ...
).
The Riemann curvature tensor is also the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
of the covariant derivative of an arbitrary covector
with itself:
:
since the
connection is torsionless, which means that the
torsion tensor vanishes.
This formula is often called the ''Ricci identity''.
This is the classical method used by
Ricci and
Levi-Civita to obtain an expression for the Riemann curvature tensor. In this way, the tensor character of the set of quantities
is proved.
This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows
:
This formula also applies to
tensor densities without alteration, because for the Levi-Civita (''not generic'') connection one gets:
:
where
:
It is sometimes convenient to also define the purely covariant version by
:
Symmetries and identities
The Riemann curvature tensor has the following symmetries and identities:
where the bracket
refers to the inner product on the tangent space induced by the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
and
the brackets and parentheses on the indices denote the
antisymmetrization and
symmetrization
In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables.
Similarly, antisymmetrization converts any function in n variables into an antisymmetric function.
Two variables
Let S ...
operators, respectively. If there is nonzero
torsion, the Bianchi identities involve the
torsion tensor.
The first (algebraic) Bianchi identity was discovered by
Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the differential
Bianchi identity.
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
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
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
contraction
Contraction may refer to:
Linguistics
* Contraction (grammar), a shortened word
* Poetic contraction, omission of letters for poetic reasons
* Elision, omission of sounds
** Syncope (phonology), omission of sounds in a word
* Synalepha, merged ...
of the first and third indices of the Riemann tensor.
:
Special cases
Surfaces
For a two-dimensional
surface, the Bianchi identities imply that the Riemann tensor has only one independent component, which means that the
Ricci scalar
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geome ...
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
is the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
and
is a function called the
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
...
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 In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a p ...
of the surface. It is also exactly half the
scalar curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geome ...
of the 2-manifold, while the
Ricci curvature tensor of the surface is simply given by
:
Space forms
A Riemannian manifold is a
space form
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consid ...
if its
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a 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
*
Introduction to the mathematics of general relativity
*
Decomposition of the Riemann curvature tensor
*
Curvature of Riemannian manifolds
*
Ricci curvature tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
Citations
References
*
*
*
*
{{DEFAULTSORT:Riemann Curvature Tensor
Bernhard Riemann
Curvature (mathematics)
Differential geometry
Riemannian geometry
Riemannian manifolds
Tensors in general relativity