Covariant derivative

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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 via a connection form.

This article presents a traditional introduction, using a coordinate system, to the covariant derivative of a vector field with respect to a vector. The covariant derivative of a tensor field is presented as an extension of the same concept. Finally, it discusses how the covariant derivative generalizes to a notion of differentiation on a vector bundle, also known as a Koszul connection.

Contents

1 Introduction and history
2 Motivation

2.1 Remarks

3 Formal definition

3.1 Functions
3.2 Vector fields
3.3 Covector fields
3.4 Tensor fields

4 Coordinate description
5 Notation
6 Derivative along curve
7 Relation to Lie derivative
8 Koszul connection

8.1 Definition
8.2 Curvature and flatness
8.3 Examples

9 Notes
10 See also

Introduction and history

Historically, at the turn of the 20th century, the covariant derivative was introduced by Gregorio Ricci-Curbastro and Tullio Levi-CivitaLevi-Civita, T. and Ricci, G. "Méthodes de calcul différential absolu et leurs applications", Math. Ann. B, 54 (1900) 125-201. in the theory of Riemannian and pseudo-Riemannian geometry. Ricci and Levi-Civita observed that the Christoffel symbols, which until that point in history had only been used to define the curvatureRiemann, G.F.B., "Über die Hypothesen, welche der Geomtrie zu Grunde liegen", Gesammelte Mathematische Werke (1866); reprint, ed. Weber, H.: Dover, New York, 1953.Christoffel, E.B., "Über die Transformation der homogenen Differentialausdrücke zweiten Grades," J. für die Reine und Angew. Math. 70 (1869), 46-70., 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 Elie Cartancf. with Cartan, E. ["Sur les variétés à connexion affine et la theorie de la relativité généralisée"], Annales, Ecole Normale 40 (1923), 325-412., that a covariant derivative could be defined in abstracto 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 away 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 form 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 Koszul, J. L. "Homologie et cohomologie des algebres de Lie", Bulletin de la Société Mathématique 78 (1950) 65-127.. 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 (and other analogous non-tensorial) 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 [\nabla_] which takes as its inputs: (1) a vector u defined at a point P, and (2) a vector field v defined in a neighborhood of PThe covariant derivative is also denoted variously by [\partial]_vu, D_vu, or other notations.. The output is then a vector [\nabla_(P)], also at the point P. The primary difference with the usual directional derivative is that [\nabla_] 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 vector transforms 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 (hence the name).

In the case of Euclidean space with an orthonormal coordinate system, 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. 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 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) = (r(t), θ(t)). A vector at a particular time tIn many applications, it may be better not to think of t as corresponding to time, at least for applications in general relativity. It is simply regarded as an abstract parameter varying smoothly and monotonically along the path. (for instance, the acceleration of the curve) is expressed in terms of [(_r, _)], where [_r] 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 (the Christoffel symbols) serve to express this change.

In a curved space, such as the surface of the Earth (regarded as a sphere), 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 (keeping it parallel to itself) drag it along a meridian to the pole N and (keeping the direction there) 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, a given metric uniquely defines a special covariant derivative called the Levi-Civita connection.

The properties of a derivative imply that [\nabla_ ] depends on the surrounding of point p in the same way as e.g. the derivative of a scalar function along a curve in a given point p depends on the surroundings of p.

The covariant derivative can be described by a "tensor" in a fixed coordinate chart, but it is not a true tensor in the sense that it is not invariant under coordinate changes.

The information on the surroundings 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.

Formal definition

A covariant derivative is a Koszul connection for the tangent bundle and other tensor bundles. Thus it has a certain behavior on functions, on vector fields, on the duals of vector fields (i.e., covector fields), and most generally of all, on arbitrary tensor fields.

Functions

Given a function [f], the covariant derivative [\nabla_f] coincides with the normal differentiation of a real function in the direction of the vector v, usually denoted by [f] and by [df()].

Vector fields

A covariant derivative [\nabla] of a vector field [] in the direction of the vector [ ] denoted [\nabla_ ] is defined by the following properties for any vector v, vector fields u, w and scalar functions f and g:

[\nabla_ ] is algebraically linear in [] so [\nabla_+g} =f\nabla_ +g\nabla_ ]
[\nabla_ ] is additive in [] so [\nabla_(+)=\nabla_ +\nabla_ ]
[\nabla_ ] obeys the product rule, i.e. [\nabla_ f=f\nabla_ +\nabla_f] where [\nabla_f] is defined above.

Note that [\nabla_ ] 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. That means that the covariant derivative is not a tensor.

Covector fields

Given a field of covectors (or one-form) [\alpha], its covariant derivative [\nabla_\alpha] can be defined using the following identity which is satisfied for all vector fields u

[\nabla_(\alpha())=(\nabla_\alpha)()+\alpha(\nabla_).]

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 [\varphi] and [\psi] are any two tensors:

[\nabla_(\varphi\otimes\psi)=(\nabla_\varphi)\otimes\psi+\varphi\otimes(\nabla_\psi),]

and if [\varphi] and [\psi] are tensor fields of the same tensor bundle then

[\nabla_(\varphi+\psi)=\nabla_\varphi+\nabla_\psi.]

The covariant derivative of a tensor field along a vector field v is again a tensor field of the same type.

Coordinate description

This section uses the Einstein summation convention

Given coordinate functions [x^i,\ i=0,1,2,...], any tangent vector can be described by its components in the basis [e_i=]. The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination Γ^ke_k. To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field e_j along e_i.

[ \nabla_[f(V)]]

where X^Hor is the horizontal lift of the vector field X, and L is the Lie derivative in the total space of P(E).

If a connection in E (or a principal bundle associated with E) is specified by means of a parallel translation along curves, then a Koszul connection can be identified with the derivative of parallel translation. Let x_t be a curve in M, and let

:[\tau_0^t:E_\rightarrow E_]

be the parallel translation in the fibres. Then

:[\nabla__0} V = \lim_ \frac)-V_}]

defines the associated Koszul connection.