Curvature of Riemannian manifolds
Encyclopedia
In mathematics
, specifically differential geometry, the infinitesimal geometry of Riemannian manifold
s 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
. 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 Differential geometry of surfaces
.
The curvature of a pseudo-Riemannian manifold
can be expressed in the same way with only slight modifications.
(or covariant differentiation)
and Lie bracket
by the following formula:
Here
is a linear transformation of the tangent space of the manifold; it is linear in each argument.
If
and 
are coordinate vector fields then 
and therefore the formula simplifies to
i.e. the curvature tensor measures noncommutativity of the covariant derivative.
The linear transformation
is also called the curvature transformation or endomorphism.
NB. There are a few books where the curvature tensor is defined with opposite sign.
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 resp., since the second means, that the R(u, v) 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 dilatations. 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
independent components.
Yet another useful identity follows from these three:
The Bianchi identity (often the second Bianchi identity)
involves the covariant derivatives:
which depends on a section 
(i.e. a 2-plane in the tangent spaces). It is the Gauss curvature
of the
-section at p; here 
-section is a locally-defined piece of surface which has the plane 
as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of 
under the exponential map
at p.
If
are two linearly independent vectors in 
then
The following formula indicates that sectional curvature describes the curvature tensor completely:
Or in a simpler formula:
gives an alternative way to describe curvature. It is used more for general vector bundle
s, and for principal bundle
s, but it works just as well for the tangent bundle with the Levi-Civita connection
. The curvature of n-dimensional Riemannian manifold is given by an antisymmetric n×n matrix
of 2-forms (or equivalently a 2-form with values in 
, the Lie algebra
of the orthogonal group
, which is the structure group of the tangent bundle of a Riemannian manifold).
Let
be a local section of orthonormal bases. Then one can define the connection form, an antisymmetric matrix of 1-forms 
which satisfy from the following identity
Then the curvature form
is defined by
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
is an n-vector of 1-forms defined by 
.
The second Bianchi identity takes form
D denotes the exterior covariant derivative
on tangent bivector
s (elements of
), 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).
however they play an important role.
It is the full trace
of the curvature tensor; given an orthonormal basis
in the tangent space at p we have
where Ric denotes 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.
Given an orthonormal basis
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
.
In dimensions 2 and 3 Weyl curvature vanishes, but if the dimension n > 3 then the second part can be non-zero.
of Riemannian manifolds. In particular, it can be used to show that if the metric is rescaled by a conformal factor of
, then the Riemann curvature tensor changes to (seen as a (0, 4) -tensor):
where
denotes the Kulkarni–Nomizu product and Hess is the Hessian.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically differential geometry, the infinitesimal geometry of Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s 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
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
. Similar notions have found applications everywhere in differential geometry.
For a more elementary discussion see the article on curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
which discusses the curvature of curves and surfaces in 2 and 3 dimensions, as well as Differential geometry of surfaces
Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....
.
The curvature of a 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...
can be expressed in the same way with only slight modifications.
The Riemann curvature tensor
The curvature of Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connectionLevi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
(or covariant differentiation)
and Lie bracketLie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
by the following formula:Here
is a linear transformation of the tangent space of the manifold; it is linear in each argument.If
and are coordinate vector fields then and therefore the formula simplifies toi.e. the curvature tensor measures noncommutativity of the covariant derivative.
The linear transformation
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
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, just because it looks similar to the Bianchi identity below. The first two should be addressed as antisymmetry and Lie algebra property resp., since the second means, that the R(u, v) 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 dilatations. 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
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 which depends on a section (i.e. a 2-plane in the tangent spaces). It is the Gauss curvatureCurvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
of the
-section at p; here -section is a locally-defined piece of surface which has the plane as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of under the exponential mapExponential map
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
at p.
If
are two linearly independent vectors in thenThe following formula indicates that sectional curvature describes the curvature tensor completely:
Or in a simpler formula:
Curvature form
The connection formConnection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms....
gives an alternative way to describe curvature. It is used more for general vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s, and for principal bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
s, but it works just as well for the tangent bundle with the Levi-Civita connection
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
. The curvature of n-dimensional Riemannian manifold is given by an antisymmetric n×n matrix
of 2-forms (or equivalently a 2-form with values in , the Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
, which is the structure group of the tangent bundle of a Riemannian manifold).Let
be a local section of orthonormal bases. Then one can define the connection form, an antisymmetric matrix of 1-forms which satisfy from the following identityThen the curvature form
is defined byThe 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
is an n-vector of 1-forms defined by .The second Bianchi identity takes form
D denotes the exterior covariant derivative
Exterior covariant derivative
In mathematics, the exterior covariant derivative, sometimes also covariant exterior derivative, is a very useful notion for calculus on manifolds, which makes it possible to simplify formulas which use a principal connection....
The curvature operator
It is sometimes convenient to think about curvature as an operatoron tangent bivector
Outer product
In linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix...
s (elements of
), 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, usually denoted by Sc.It is the full trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
of the curvature tensor; given an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
in the tangent space at p we havewhere Ric denotes 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 Ric.Given an orthonormal basis
in the tangent space at p we haveThe 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
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
is given in the article on Christoffel symbols
Christoffel symbols
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
.
Weyl curvature tensor
The Weyl curvature tensor has the same symmetries as the curvature tensor, plus one extra: its Ricci curvature must vanish.In dimensions 2 and 3 Weyl curvature vanishes, but if the dimension n > 3 then the second part can be non-zero.
- The curvature tensor can be decomposed into the part which depends on the Ricci curvature, and the Weyl tensor.
- If g′=fg for some positive scalar function f — a conformalConformal mapIn mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...
change of metric — then W ′ = W. - For a manifoldManifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
of constant curvatureConstant curvatureIn mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points...
, the Weyl tensor is zero.- Moreover, W=0 if and only if the metric is locally conformalConformal mapIn mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...
to the standard Euclidean metric (equal to fg, where g is the standard metric in some coordinate frame and f is some scalar function).
- Moreover, W=0 if and only if the metric is locally conformal
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 geometryConformal geometry
In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...
of Riemannian manifolds. In particular, it can be used to show that if the metric is rescaled by a conformal factor of
, then the Riemann curvature tensor changes to (seen as a (0, 4) -tensor):
where
denotes the Kulkarni–Nomizu product and Hess is the Hessian.Calculation of curvature
For calculation of curvature- of hypersurfaces and submanifolds see second fundamental form,
- in coordinates see the list of formulas in Riemannian geometry or covariant derivativeCovariant derivativeIn 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 given...
, - by moving frames see Cartan connectionCartan connectionIn the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the...
and curvature formCurvature formIn differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...
. - the Jacobi equation can help if one knows something about the behavior of geodesics.