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Encyclopedia > 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 differential operator, to be contrasted with the approach via a connection form. For other meanings of mathematics or math, see mathematics (disambiguation). ... In mathematics, a derivative is the rate of change of a quantity. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ...


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. In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

Contents

Introduction and history

Historically, at the turn of the 20th century, the covariant derivative was introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita[1] 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 curvature[2][3], 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. (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999... Gregorio Ricci-Curbastro (January 12, 1853 - August 6, 1925) was an Italian mathematician. ... Tullio Levi-Civita. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ... In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In category theory, see covariant functor. ...


It was soon noted by other mathematicians, prominent among these being Hermann Weyl, Jan Arnoldus Schouten, and Elie Cartan[4], 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. Hermann Weyl Hermann Weyl (November 9, 1885 – December 8, 1955) was a German mathematician. ... Jan Arnoldus Schouten (August 28, 1883 - January 20, 1971) was a Dutch mathematician. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...


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 [5]. 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. The 1940s decade ran from 1940 to 1949. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... For more technical Wiki articles on tensors, see the section later in this article. ... In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... 1950 (MCML) was a common year starting on Sunday (link will take you to calendar). ... Jean-Louis Koszul (born January 3, 1920 in Strasbourg, France) is a mathematician best known for studying geometry and discovering the Koszul complex. ... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... Lie algebra cohomology is a cohomology theory for Lie algebras. ... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...


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_{bold u}{bold v} 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 P[6]. The output is then a vector nabla_{bold u}{bold v}(P), also at the point P. The primary difference with the usual directional derivative is that nabla_{bold u}{bold v} must, in a certain precise sense, be independent of the manner in which it is expressed in a coordinate system. In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...


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 linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In linear algebra, we may consider some finite-dimensional vector space, which can have associated with it some basis with which we can work with respect to. ... It has been suggested that this article or section be merged with Covariant. ...


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. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...


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. This article describes some of the common coordinate systems that appear in elementary mathematics. ...


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 t[7] (for instance, the acceleration of the curve) is expressed in terms of ({mathbf e}_r, {mathbf e}_{theta}), where {mathbf e}_r and {mathbf e}_{theta} 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 mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...


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. In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...


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.
In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...


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_{mathbf v} {mathbf u} 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.

In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ... In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ... // Mathmatics In mathematics, the term torsion has several meanings, mostly unrelated to each other. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ...

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. In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...


Functions

Given a function f, the covariant derivative nabla_{mathbf v}f coincides with the normal differentiation of a real function in the direction of the vector v, usually denoted by {mathbf v}f and by df({mathbf v}).


Vector fields

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

  1. nabla_{mathbf v} {mathbf u} is algebraically linear in {mathbf v} so nabla_{f{mathbf v}+g{mathbf w}} {mathbf u}=fnabla_{mathbf v} {mathbf u}+gnabla_{mathbf w} {mathbf u}
  2. nabla_{mathbf v} {mathbf u} is additive in {mathbf u} so nabla_{mathbf v}({mathbf u}+{mathbf w})=nabla_{mathbf v} {mathbf u}+nabla_{mathbf v} {mathbf w}
  3. nabla_{mathbf v} {mathbf u} obeys the product rule, i.e. nabla_{mathbf v} f{mathbf u}=fnabla_{mathbf v} {mathbf u}+{mathbf u}nabla_{mathbf v}f where nabla_{mathbf v}f is defined above.

Note that nabla_{mathbf v} {mathbf u} 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. In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...


Covector fields

Given a field of covectors (or one-form) α, its covariant derivative nabla_{mathbf v}alpha can be defined using the following identity which is satisfied for all vector fields u In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... A one-form, also called a covector, is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space. ...

nabla_{mathbf v}(alpha({mathbf u}))=(nabla_{mathbf v}alpha)({mathbf u})+alpha(nabla_{mathbf v}{mathbf 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 varphi and ψ are any two tensors: In mathematics, the modern component-free approach to the theory of tensors views tensors initially as abstract objects, expressing some definite type of multi-linear concept. ...

nabla_{mathbf v}(varphiotimespsi)=(nabla_{mathbf v}varphi)otimespsi+varphiotimes(nabla_{mathbf v}psi),

and if varphi and ψ are tensor fields of the same tensor bundle then

nabla_{mathbf v}(varphi+psi)=nabla_{mathbf v}varphi+nabla_{mathbf v}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={partialoverpartial x^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 Γkek. To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field ej along ei. For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ... In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...

nabla_{{mathbf e}_i} {mathbf e}_j = Gamma^k {}_{i j} {mathbf e}_k,

the coefficients Γki j are called Christoffel symbols. Then using the rules in the definition, we find that for general vector fields {mathbf v}= v^ie_i and {mathbf u}= u^ie_i we get In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

nabla_{mathbf v} {mathbf u} = (v^i u^j Gamma^k {}_{i j}+v^i{partial u^koverpartial x^i}){mathbf e}_k,

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

nabla_{{mathbf e}_j} {mathbf u}=nabla_j {mathbf u} = left( frac{partial u^i}{partial x^j} + u^k Gamma^i {}_{jk} right) {mathbf e}_i

In words: the covariant derivative is the normal derivative along the coordinates along with correction terms which tell how the coordinates change.


The covariant derivative of a type (r,s) tensor field along ec is given by the expression:

(nabla_c T)^{a_1 ldots a_r}{}_{b_1 ldots b_s} = frac{partial}{partial x^c}T^{a_1 ldots a_r}{}_{b_1 ldots b_s}+,Gamma ^{a_1}{}_{dc} T ^{d ldots a_r}{}_{b_1 ldots b_s} + ldots + Gamma ^{a_r}{}_{dc} T ^{a_1 ldots a_{r-1}d}{}_{b_1 ldots b_s}
-,Gamma ^d {}_{b_1 c} T ^{a_1 ldots a_r}{}_{d ldots b_s} - ldots - Gamma ^d {}_{b_s c} T ^{a_1 ldots a_r}{}_{b_1 ldots b_{s-1} d}.

Or, in words: take the partial derivative of the tensor and add: a +Gamma^{a_i}{}_{dc} for every upper index ai, and a -Gamma^{d}{}_{b_ic} for every lower index bi.


If instead of a tensor, one is trying to differentiate a tensor density (of weight +1), then you also add a term Tensor field - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...

-Gamma^d{}_{d c} T^{a_1 ldots a_r}{}_{b_1 ldots b_s}.

If it is a tensor density of weight W, then multiply that term by W. For example, sqrt{-g} is a scalar density (of weight +1), so we get:

(sqrt{-g})_{;c} = (sqrt{-g})_{,c} - sqrt{-g},Gamma^{d}{}_{d c}

where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. By the way, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.


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: A semicolon ( ; ) is a punctuation mark. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... The term comma has various uses; comma is the name used for one of the punctuation symbols: , The term comma is also used in music theory for various small intervals that arise as differences between approximately equal intervals. ...

nabla_{e_j} {mathbf v} stackrel{mathrm{def}}{=} v^s {}_{;j}e_s ;;;;;; v^i {}_{;j} = v^i {}_{,j} + v^kGamma^i {}_{k j}

Once again this shows that the covariant derivative of a vector field is not just simply obtained by differentiating to the coordinates vi,j, but also depends on the vector v itself through vkΓikj.


In some older texts (notably Adler, Bazin & Schiffer, Introduction to General Relativity), the covariant derivative is denoted by a double pipe:

nabla_j {mathbf v} stackrel{mathrm{def}}{=} v^i {}_{||j} ;;;;;;

Derivative along curve

Since the covariant derivative nabla_XT depends only on value of X at the point one can define covariant derivative along a smooth curve γ(t) in manifold:

D_tT=nabla_{dotgamma(t)}T.

In this case tensor field T has to be defined only on the curve γ(t).


In particular, dot{gamma}(t) is a vector field along the curve γ itself. If nabla_{dotgamma(t)}dotgamma(t) vanishes, the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metrics, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to the arc length. In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...


Derivative along a curve is also used to define the parallel transport along the same curve. In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...


Sometimes 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 (over C∞(M)) in the direction argument, while the Lie derivative is linear in neither argument. In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...


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. Torsion of affine connection is a (1,2) tensor given by the formula where is the Lie bracket of the two vector fields. ...


Koszul connection

Definition

Let E be a vector bundle over a smooth manifold M, and let TM be the tangent bundle of M. Denote the space of sections of E by Γ(E) and that of TM by Γ(TM). A Koszul connection on E is an R-bilinear mapping In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

Gamma(TM)timesGamma(E)rightarrowGamma(E),quad (V,X)mapstonabla_V X

such that the following conditions are satisfied:

  1. nabla_{fV}X = fnabla_V X for all f ∈ C∞(M),
  2. nabla_V(fX) = (L_Vf)X+fnabla_VX for all f ∈ C∞(M), where LV is the Lie derivative of functions along vector fields, and
  3. nabla_V(X+Y) = nabla_VX+nabla_VY.

In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

Curvature and flatness

The curvature of the Koszul connection (as for any connection) is defined by

R(V,W) = nabla_Vnabla_W-nabla_Wnabla_V-nabla_{[V,W]}.

One advantage of defining the connection in this way is that the following theorem becomes a statement about derivations on Lie algebras, to which one may then apply purely algebraic techniques from Lie algebra cohomology:

There exists a local trivialization of the bundle E with a basis of parallel sections if, and only if, the curvature vanishes identically.

In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...

Examples

  • The classical covariant derivative is a Koszul connection on the tangent bundle of M, or more generally on any tensor bundle formed by taking tensor products of the tangent bundle with itself and its dual.
  • If P(E) → M is the frame bundle, consisting of all bases of E under the action of G = GL(r) where r is the rank of E, then an Ehresmann connection on P(E) induces a Koszul connection on E as follows. There is a natural one-to-one correspondence between (local) sections of E and functions φ : P(E) → Rr (defined locally over the base) which are equivariant under the action of G. Let the function φ associated to V be denoted f(V) for each local section V of E. The associated Koszul connection may be defined by
f(nabla_X V) = L_{X^{Hor}}[f(V)]
where XHor 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 xt be a curve in M, and let
tau_0^t:E_{x_t}rightarrow E_{x_0}
be the parallel translation in the fibres. Then
nabla_{dot{x}_0} V = lim_{trightarrow 0} frac{tau_0^t(V_{x_t})-V_{x_0}}{t}
defines the associated Koszul connection.

In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... In differential geometry, an Ehresmann connection is a version of the connection concept which applies to arbitrary fibre bundles. ... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

Notes

  1. ^ Levi-Civita, T. and Ricci, G. "Méthodes de calcul différential absolu et leurs applications", Math. Ann. B, 54 (1900) 125-201.
  2. ^ Riemann, G.F.B., "Über die Hypothesen, welche der Geomtrie zu Grunde liegen", Gesammelte Mathematische Werke (1866); reprint, ed. Weber, H.: Dover, New York, 1953.
  3. ^ Christoffel, E.B., "Über die Transformation der homogenen Differentialausdrücke zweiten Grades," J. für die Reine und Angew. Math. 70 (1869), 46-70.
  4. ^ cf. with Cartan, E. "Sur les variétés à connexion affine et la theorie de la relativité généralisée", Annales, Ecole Normale 40 (1923), 325-412.
  5. ^ Koszul, J. L. "Homologie et cohomologie des algebres de Lie", Bulletin de la Société Mathématique 78 (1950) 65-127.
  6. ^ The covariant derivative is also denoted variously by partialvu, Dvu, or other notations.
  7. ^ In 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.

Bernhard Riemann. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

See also

  1. Basic introduction to the mathematics of curved spacetime
  2. Connection
  3. Connection form
  4. Levi-Civita connection
  5. Christoffel symbols
  6. Gauge covariant derivative
  7. Parallel transport
  8. exterior covariant derivative

  Results from FactBites:
 
Covariant derivative - Wikipedia, the free encyclopedia (1228 words)
In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
The covariant derivative can be described by a tensor in a fixed coordinate chart, but it is not a tensor in the sense that it is not invariant under coordinate changes.
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.
PlanetMath: connection (2998 words)
Since the notions of connection, parallel transport, and covariant derivative are so closely related, it is easy to translate propositions involving one of these terms into propsitions involving a different one of three terms.
This property is the origin of the term ``covariant derivative'' -- the covariant derivative maps tensor fields into quantities which transform in the same manner.
Sometimes, as in the theory of embedded surfaces, there are two connections present so a semicolon is used to indicate covariant derivatives with repsect to one connection and a vertical bar or a colon is used to indicate covariant derivatives with respect to the other connection.
  More results at FactBites »


 

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