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 given by a principal connection on the frame bundle – see affine
connection. In the special case of a manifold isometrically embedded into a higher-dimensional
Euclidean space, the covariant derivative can be viewed as the orthogonal projection
of the Euclidean derivative along a tangent vector onto the manifold's tangent space.
In this case the Euclidean derivative is broken into two parts, the extrinsic normal component
and the intrinsic covariant derivative component. This article presents an introduction to the
covariant derivative of a vector field with respect to a vector field, both in a coordinate
free language and using a local coordinate system and the traditional index notation.
The covariant derivative of a tensor field is presented as an extension of the same concept.
The covariant derivative generalizes straightforwardly to a notion of differentiation associated
to a connection on a vector bundle, also known as a Koszul connection.
Introduction and history Historically, at the turn of the 20th century,
the covariant derivative was introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita
in the theory of Riemannian and pseudo-Riemannian geometry. Ricci and Levi-Civita observed that
the Christoffel symbols used to define the curvature could also provide a notion of differentiation
which generalized the classical directional derivative of vector fields on a manifold.
This new derivative – the Levi-Civita connection – was covariant in the sense that it satisfied
Riemann's requirement that objects in geometry should be independent of their description
in a particular coordinate system. It was soon noted by other mathematicians,
prominent among these being Hermann Weyl, Jan Arnoldus Schouten, and Élie Cartan, that
a covariant derivative could be defined abstractly without the presence of a metric. The crucial
feature was not a particular dependence on the metric, but that the Christoffel symbols
satisfied a certain precise second order transformation law. This transformation law could serve as
a starting point for defining the derivative in a covariant manner. Thus the theory of
covariant differentiation forked off from the strictly Riemannian context to include
a wider range of possible geometries. In the 1940s, practitioners of differential
geometry began introducing other notions of covariant differentiation in general vector
bundles which were, in contrast to the classical bundles of interest to geometers, not part
of the tensor analysis of the manifold. By and large, these generalized covariant derivatives
had to be specified ad hoc by some version of the connection concept. In 1950, Jean-Louis
Koszul unified these new ideas of covariant differentiation in a vector bundle by means
of what is known today as a Koszul connection or a connection on a vector bundle. Using
ideas from Lie algebra cohomology, Koszul successfully converted many of the analytic
features of covariant differentiation into algebraic ones. In particular, Koszul connections
eliminated the need for awkward manipulations of Christoffel symbols objects in differential
geometry. Thus they quickly supplanted the classical notion of covariant derivative in
many post-1950 treatments of the subject. Motivation
The covariant derivative is a generalization of the directional derivative from vector
calculus. As with the directional derivative, the covariant derivative is a rule, , which
takes as its inputs: a vector, u, defined at a point P, and a vector field, v, defined
in a neighborhood of P. The output is the vector , also at the point P. The primary
difference from the usual directional derivative is that must, in a certain precise sense,
be independent of the manner in which it is expressed in a coordinate system.
A vector may be described as a list of numbers in terms of a basis, but as a geometrical
object a vector retains its own identity regardless of how one chooses to describe it in a basis.
This persistence of identity is reflected in the fact that when a vector is written
in one basis, and then the basis is changed, the components of the vector transform according
to a change of basis formula. Such a transformation law is known as a covariant transformation.
The covariant derivative is required to transform, under a change in coordinates, in the same
way as a vector does: the covariant derivative must change by a covariant transformation.
In the case of Euclidean space, one tends to define the derivative of a vector field
in terms of the difference between two vectors at two nearby points. In such a system one
translates one of the vectors to the origin of the other, keeping it parallel. With a
Cartesian coordinate system we thus obtain the simplest example: covariant derivative
which is obtained by taking the derivative of the components.
In the general case, however, one must take into account the change of the coordinate
system. For example, if the same covariant derivative is written in polar coordinates
in a two dimensional Euclidean plane, then it contains extra terms that describe how
the coordinate grid itself "rotates". In other cases the extra terms describe how the coordinate
grid expands, contracts, twists, interweaves, etc.
Consider the example of moving along a curve γ(t) in the Euclidean plane. In polar coordinates,
γ may be written in terms of its radial and angular coordinates by γ(t) =, θ(t)). A
vector at a particular time t is expressed in terms of , where and are unit tangent vectors
for the polar coordinates, serving as a basis to decompose a vector in terms of radial and
tangential components. At a slightly later time, the new basis in polar coordinates appears
slightly rotated with respect to the first set. The covariant derivative of the basis
vectors serve to express this change.
In a curved space, such as the surface of the Earth, the translation is not well defined
and its analog, parallel transport, depends on the path along which the vector is translated.
A vector e on a globe on the equator in Q is directed to the north. Suppose we parallel
transport the vector first along the equator until P and then drag it along a meridian
to the pole N and subsequently transport it along another meridian back to Q. Then we
notice that the parallel-transported vector along a closed circuit does not return as
the same vector; instead, it has another orientation. This would not happen in Euclidean space and
is caused by the curvature of the surface of the globe. The same effect can be noticed
if we drag the vector along an infinitesimally small closed surface subsequently along two
directions and then back. The infinitesimal change of the vector is a measure of the curvature.
Remarks The definition of the covariant derivative
does not use the metric in space. However, for each metric there is a unique torsion-free
covariant derivative called the Levi-Civita connection such that the covariant derivative
of the metric is zero. The properties of a derivative imply that
depends on an arbitrarily small neighborhood of a point p in the same way as e.g. the derivative
of a scalar function along a curve at a given point p depends on an arbitrarily small neighborhood
of p. The information on the neighborhood of a point
p in the covariant derivative can be used to define parallel transport of a vector.
Also the curvature, torsion, and geodesics may be defined only in terms of the covariant
derivative or other related variation on the idea of a linear connection.
Informal definition using an embedding into Euclidean space
Assume a Riemann manifold is embedded into Euclidean space via a differentiable mapping
such that the tangent space at is spanned by the vectors
and the scalar product on is compatible with the metric on M: .
For a tangent vector field one has .
The last term is not tangential to M, but can be expressed as a linear combination of
the tangent space base vectors using the Christoffel symbols as linear factors plus a non-tangent
vector: .
The covariant derivative is defined as just a tangential portion of the usual derivative:
In the case of the Levi-Civita connection is required to be orthogonal to tangent space,
so .
On the other hand
implies
and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric:
Formal definition A covariant derivative is a connection on
the tangent bundle and other tensor bundles. Thus it has a certain behavior on functions,
on vector fields, on the duals of vector fields, and most generally of all, on arbitrary tensor
fields. Functions
Given a function , the covariant derivative coincides with the normal differentiation
of a real function in the direction of the vector v, usually denoted by and by .
Vector fields A covariant derivative of a vector field in
the direction of the vector denoted is a new vector field defined by the following properties:
is algebraically linear in so is additive in so
obeys the product rule, i.e. where is defined above.
Note that at point p depends on the value of v at p and on values of u in a neighbourhood
of p because of the last property, the product rule.
Covector fields Given a field of covectors , its covariant
derivative can be defined using the following identity which is satisfied for all vector
fields u
The covariant derivative of a covector field along a vector field v is again a covector
field. Tensor fields
Once the covariant derivative is defined for fields of vectors and covectors it can be
defined for arbitrary tensor fields using the following identities where and are any
two tensors:
and if and are tensor fields of the same tensor bundle then
The covariant derivative of a tensor field along a vector field v is again a tensor field
of the same type. Explicitly, let T be a tensor field of type.
Consider T to be a differentiable multilinear map of smooth sections α1, α2, ..., αq
of the cotangent bundle T*M and of sections X1, X2, ... Xp of the tangent bundle TM, written
T(α1, α2, ..., X1, X2, ...) into R. The covariant derivative of T along Y is given
by the formula
Coordinate description
Given coordinate functions ,
any tangent vector can be described by its components in the basis
. The covariant derivative of a basis vector
along a basis vector is again a vector and so can be expressed as a linear combination
. To specify the covariant derivative it is enough to specify the covariant derivative
of each basis vector field along .
the coefficients are called Christoffel symbols. Then using the rules in the definition, we
find that for general vector fields and we get
so
The first term in this formula is responsible for "twisting" the coordinate system with
respect to the covariant derivative and the second for changes of components of the vector
field u. In particular
In words: the covariant derivative is the usual derivative along the coordinates with
correction terms which tell how the coordinates change.
The covariant derivative of a type tensor field along is given by the expression:
Or, in words: take the partial derivative of the tensor and add: a for every upper index
, and a for every lower index . If instead of a tensor, one is trying to differentiate
a tensor density, then you also add a term
If it is a tensor density of weight W, then multiply that term by W. For example, is a
scalar density, so we get:
where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation.
Incidentally, this particular expression is equal to zero, because the covariant derivative
of a function solely of the metric is always zero.
Examples For a scalar field , covariant differentiation
is simply partial differentiation:
For a contravariant vector field , we have:
For a covariant vector field , we have:
For a type tensor field , we have:
The notation above is meant in the sense
One must always remember that covariant derivatives do not commute, i.e. . It is actually easy
to show that:
where is the Riemann tensor. Similarly,
and
The latter can be shown by taking that . Notation
In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its
components in this equation. Often a notation is used in which the covariant
derivative is given with a semicolon, while a normal partial derivative is indicated by
a comma. In this notation we write the same as:
Once again this shows that the covariant derivative of a vector field is not just simply obtained
by differentiating to the coordinates , but also depends on the vector v itself through
. In some older texts, the covariant derivative
is denoted by a double pipe:
Derivative along curve Since the covariant derivative of a tensor
field at a point depends only on value of the vector field at one can define the covariant
derivative along a smooth curve in a manifold:
Note that the tensor field only needs to be defined on the curve for this definition to
make sense. In particular, is a vector field along the
curve itself. If vanishes then the curve is called a geodesic of the covariant derivative.
If the covariant derivative is the Levi-Civita connection of a certain metric then the geodesics
for the connection are precisely the geodesics of the metric that are parametrised by arc
length. The derivative along a curve is also used
to define the parallel transport along the curve.
Sometimes the covariant derivative along a curve is called absolute or intrinsic derivative.
Relation to Lie derivative A covariant derivative introduces an extra
geometric structure on a manifold which allows vectors in neighboring tangent spaces to be
compared. This extra structure is necessary because there is no canonical way to compare
vectors from different vector spaces, as is necessary for this generalization of the directional
derivative. There is however another generalization of directional derivatives which is canonical:
the Lie derivative. The Lie derivative evaluates the change of one vector field along the flow
of another vector field. Thus, one must know both vector fields in an open neighborhood.
The covariant derivative on the other hand introduces its own change for vectors in a
given direction, and it only depends on the vector direction at a single point, rather
than a vector field in an open neighborhood of a point. In other words, the covariant
derivative is linear) in the direction argument, while the Lie derivative is linear in neither
argument. Note that the antisymmetrized covariant derivative
∇uv − ∇vu, and the Lie derivative Luv differ by the torsion of the connection, so
that if a connection is symmetric, then its antisymmetrization is the Lie derivative.
See also
Notes
References Kobayashi, Shoshichi and Nomizu, Katsumi.
Foundations of Differential Geometry, Vol. 1. Wiley Interscience. ISBN 0-471-15733-3.
I.Kh. Sabitov, "Covariant differentiation", in Hazewinkel, Michiel, Encyclopedia of Mathematics,
Springer, ISBN 978-1-55608-010-4 Sternberg, Shlomo. Lectures on Differential
Geometry. Prentice-Hall. Spivak, Michael. A Comprehensive Introduction
to Differential Geometry. Publish or Perish, Inc.