Derivado covariante

Specification of derivatives along tangent vectors of a manifold

Este artículo trata sobre derivados covariantes. Para las derivadas del tensor direccional con respecto a la mecánica del continuo, consulte Derivada del tensor (mecánica del continuo) . Para conocer la derivada covariante utilizada en las teorías de calibre , consulte Derivada covariante de calibre .

En matemáticas , la derivada covariante es una forma de especificar una derivada a lo largo de los vectores tangentes de una variedad . Alternativamente, la derivada covariante es una forma de introducir y trabajar con una conexión en un colector por medio de un operador diferencial , para contrastar con el enfoque dado por una conexión principal en el paquete de tramas - ver conexión afín . En el caso especial de una variedad incrustada isométricamente en un espacio euclidiano de dimensiones superiores , la derivada covariante puede verse como la proyección ortogonal de la euclidiana.derivada direccional en el espacio tangente de la variedad. En este caso, el derivado euclidiano se divide en dos partes, el componente normal extrínseco (que depende de la incrustación) y el componente derivado covariante intrínseco.

El nombre está motivado por la importancia de los cambios de coordenadas en la física : la derivada covariante se transforma de forma covariante bajo una transformación de coordenadas general, es decir, linealmente a través de la matriz jacobiana de la transformación. ^[1]

Este artículo presenta una introducción a la derivada covariante de un campo vectorial con respecto a un campo vectorial, tanto en un lenguaje sin coordenadas como utilizando un sistema de coordenadas local y la notación de índice tradicional. La derivada covariante de un campo tensorial se presenta como una extensión del mismo concepto. La derivada covariante se generaliza directamente a una noción de diferenciación asociada a una conexión en un paquete de vectores , también conocida como conexión Koszul .

Historia

Históricamente, a principios del siglo XX, la derivada covariante fue introducida por Gregorio Ricci-Curbastro y Tullio Levi-Civita en la teoría de la geometría riemanniana y pseudo-riemanniana . ^[2] Ricci y Levi-Civita (siguiendo las ideas de Elwin Bruno Christoffel ) observaron que los símbolos de Christoffel utilizados para definir la curvatura también podían proporcionar una noción de diferenciación que generalizaba la derivada direccional clásica de los campos vectoriales en una variedad. ^{[3] [4]} Este nuevo derivado - elConexión Levi-Civita : era covariante en el sentido de que satisfacía el requisito de Riemann de que los objetos en geometría debían ser independientes de su descripción en un sistema de coordenadas particular.

Pronto fue notado por otros matemáticos, entre los que se destacan Hermann Weyl , Jan Arnoldus Schouten y Élie Cartan , ^[5] que una derivada covariante podría definirse de manera abstracta sin la presencia de una métrica . La característica crucial no era una dependencia particular de la métrica, sino que los símbolos de Christoffel cumplían una cierta ley precisa de transformación de segundo orden. Esta ley de transformación podría servir como punto de partida para definir la derivada de manera covariante. Así, la teoría de la diferenciación covariante se bifurcó del contexto estrictamente riemanniano para incluir una gama más amplia de geometrías posibles.

En la década de 1940, los practicantes de la geometría diferencial comenzaron a introducir otras nociones de diferenciación covariante en paquetes vectoriales generales que, en contraste con los paquetes clásicos de interés para los geómetras, no formaban parte del análisis tensorial de la variedad. En general, estas derivadas covariantes generalizadas tenían que ser especificadas ad hoc por alguna versión del concepto de conexión. En 1950, Jean-Louis Koszul unificó estas nuevas ideas de diferenciación covariante en un paquete vectorial mediante lo que hoy se conoce como una conexión Koszul o una conexión en un paquete vectorial. ^[6] Usando ideas de la cohomología del álgebra de LieKoszul convirtió con éxito muchas de las características analíticas de la diferenciación covariante en algebraicas. En particular, las conexiones de Koszul eliminaron la necesidad de manipulaciones incómodas de los símbolos de Christoffel (y otros objetos análogos no tensoriales ) en geometría diferencial. Así, rápidamente suplantaron la noción clásica de derivada covariante en muchos tratamientos del tema posteriores a 1950.

Motivación

La derivada covariante es una generalización de la derivada direccional del cálculo vectorial . Al igual que con la derivada direccional, la derivada covariante es una regla, que toma como entradas: (1) un vector, u , definido en un punto P , y (2) un campo vectorial , v , definido en una vecindad de P . ^[7] La salida es el vector , también en el punto P . La principal diferencia con la derivada direccional habitual es que debe, en cierto sentido preciso, ser independiente de la manera en que se expresa en un${\displaystyle \nabla _{\mathbf {u} }{\mathbf {v} }}$${\displaystyle \nabla _{\mathbf {u} }{\mathbf {v} }(P)}$${\displaystyle \nabla _{\mathbf {u} }{\mathbf {v} }}$sistema de coordenadas .

Un vector puede describirse como una lista de números en términos de una base , pero como un objeto geométrico, un vector retiene su propia identidad independientemente de cómo se elija describirlo en una base. Esta persistencia de la identidad se refleja en el hecho de que cuando un vector se escribe en una base, y luego se cambia la base, los componentes del vector se transforman de acuerdo con una fórmula de cambio de base . Esta ley de transformación se conoce como transformación covariante . Se requiere que la derivada covariante se transforme, bajo un cambio de coordenadas, de la misma manera que lo hace una base: la derivada covariante debe cambiar por una transformación covariante (de ahí el nombre).

En el caso del espacio euclidiano , se tiende a definir la derivada de un campo vectorial en términos de la diferencia entre dos vectores en dos puntos cercanos. En tal sistema, uno traslada uno de los vectores al origen del otro, manteniéndolo paralelo. Con un sistema de coordenadas cartesiano ( ortonormal fijo ), "mantenerlo paralelo" equivale a mantener constantes los componentes. El espacio euclidiano proporciona el ejemplo más simple: una derivada covariante que se obtiene tomando la derivada direccional ordinaria de los componentes en la dirección del vector de desplazamiento entre los dos puntos cercanos.

En el caso general, sin embargo, se debe tener en cuenta el cambio del sistema de coordenadas. Por ejemplo, si la misma derivada covariante se escribe en coordenadas polares en un plano euclidiano bidimensional, entonces contiene términos adicionales que describen cómo la propia cuadrícula de coordenadas "gira". En otros casos, los términos adicionales describen cómo la cuadrícula de coordenadas se expande, contrae, retuerce, entrelaza, etc. En este caso, "mantenerla paralela" no equivale a mantener los componentes constantes durante la traslación.

Considere el ejemplo de moverse a lo largo de una curva γ ( t ) en el plano euclidiano. En coordenadas polares, γ puede escribirse en términos de sus coordenadas radiales y angulares por γ ( t ) = ( r ( t ), θ ( t )). Un vector en un tiempo particular t ^[8] (por ejemplo, la aceleración de la curva) se expresa en términos de , donde y son vectores unitarios tangentes para las coordenadas polares, que sirven como base para descomponer un vector en términos de radial y componentes tangenciales${\displaystyle (\mathbf {e} _{r},\mathbf {e} _{\theta })}$${\displaystyle \mathbf {e} _{r}}$${\displaystyle \mathbf {e} _{\theta }}$. En un momento ligeramente posterior, la nueva base en coordenadas polares aparece ligeramente rotada con respecto al primer conjunto. La derivada covariante de los vectores base (los símbolos de Christoffel ) sirve para expresar este cambio.

En un espacio curvo, como la superficie de la Tierra (considerada como una esfera), la traslación no está bien definida y su transporte paralelo análogo depende de la trayectoria por la que se traslade el vector.

Un vector e en un globo terráqueo en el ecuador en el punto Q se dirige hacia el norte. Supongamos que transportamos en paralelo el vector primero a lo largo del ecuador hasta el punto P y luego (manteniéndolo paralelo a sí mismo) lo arrastramos a lo largo de un meridiano hasta el polo N y (manteniendo la dirección allí) posteriormente lo transportamos por otro meridiano de regreso a Q. notamos que el vector transportado en paralelo a lo largo de un circuito cerrado no regresa como el mismo vector; en cambio, tiene otra orientación. Esto no sucedería en el espacio euclidiano y es causado por la curvatura 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 ${\displaystyle \nabla _{\mathbf {v} }\mathbf {u} }$ depends on the values of u on an arbitrarily small neighborhood of a point p in the same way as e.g. the derivative of a scalar function f along a curve at a given point p depends on the values of f in 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

Suppose an open subset ${\displaystyle U}$ of a Riemannian manifold ${\displaystyle M}$ is embedded into Euclidean space ${\displaystyle (\mathbb {R} ^{n},\langle \cdot ,\cdot \rangle )}$ via a twice continuously-differentiable (C²) mapping ${\displaystyle {\vec {\Psi }}:\mathbb {R} ^{d}\supset U\to \mathbb {R} ^{n}}$ such that the tangent space at ${\displaystyle {\vec {\Psi }}(p)\in M}$ is spanned by the vectors

${\displaystyle \left\lbrace \left.{\frac {\partial {\vec {\Psi }}}{\partial x^{i}}}\right|_{p}:i\in \lbrace 1,\dots ,d\rbrace \right\rbrace }$

and the scalar product ${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$ on ${\displaystyle \mathbb {R} ^{n}}$ is compatible with the metric on M:

${\displaystyle g_{ij}=\left\langle {\frac {\partial {\vec {\Psi }}}{\partial x^{i}}},{\frac {\partial {\vec {\Psi }}}{\partial x^{j}}}\right\rangle .}$

(Since the manifold metric is always assumed to be regular, the compatibility condition implies linear independence of the partial derivative tangent vectors.)

For a tangent vector field, ${\displaystyle {\vec {V}}=v^{j}{\frac {\partial {\vec {\Psi }}}{\partial x^{j}}}}$, one has

${\displaystyle {\frac {\partial {\vec {V}}}{\partial x^{i}}}={\frac {\partial }{\partial x^{i}}}\left(v^{j}{\frac {\partial {\vec {\Psi }}}{\partial x^{j}}}\right)={\frac {\partial v^{j}}{\partial x^{i}}}{\frac {\partial {\vec {\Psi }}}{\partial x^{j}}}+v^{j}{\frac {\partial ^{2}{\vec {\Psi }}}{\partial x^{i}\,\partial x^{j}}}.}$

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 vector orthogonal to the tangent space:

${\displaystyle {\frac {\partial ^{2}{\vec {\Psi }}}{\partial x^{i}\,\partial x^{j}}}={\Gamma ^{k}}_{ij}{\frac {\partial {\vec {\Psi }}}{\partial x^{k}}}+{\vec {n}}.}$

In the case of the Levi-Civita connection, the covariant derivative ${\displaystyle \nabla _{\mathbf {e} _{i}}{\vec {V}}}$, also written ${\displaystyle \nabla _{i}{\vec {V}}}$, is defined as the orthogonal projection of the usual derivative onto tangent space:

${\displaystyle \nabla _{\mathbf {e} _{i}}{\vec {V}}:={\frac {\partial {\vec {V}}}{\partial x^{i}}}-{\vec {n}}=\left({\frac {\partial v^{k}}{\partial x^{i}}}+v^{j}{\Gamma ^{k}}_{ij}\right){\frac {\partial {\vec {\Psi }}}{\partial x^{k}}}.}$

To obtain the relation between Christoffel symbols for the Levi-Civita connection and the metric, first we must note that, since ${\displaystyle {\vec {n}}}$ in previous equation is orthogonal to tangent space:

${\displaystyle \left\langle {\frac {\partial ^{2}{\vec {\Psi }}}{\partial x^{i}\,\partial x^{j}}},{\frac {\partial {\vec {\Psi }}}{\partial x^{l}}}\right\rangle =\left\langle {\Gamma ^{k}}_{ij}{\frac {\partial {\vec {\Psi }}}{\partial x^{k}}}+{\vec {n}},{\frac {\partial {\vec {\Psi }}}{\partial x^{l}}}\right\rangle ={\Gamma ^{k}}_{ij}\left\langle {\frac {\partial {\vec {\Psi }}}{\partial x^{k}}},{\frac {\partial {\vec {\Psi }}}{\partial x^{l}}}\right\rangle ={\Gamma ^{k}}_{ij}\,g_{kl}.}$

Second, the partial derivative of a component of the metric is:

${\displaystyle {\frac {\partial g_{ab}}{\partial x^{c}}}={\frac {\partial }{\partial x^{c}}}\left\langle {\frac {\partial {\vec {\Psi }}}{\partial x^{a}}},{\frac {\partial {\vec {\Psi }}}{\partial x^{b}}}\right\rangle =\left\langle {\frac {\partial ^{2}{\vec {\Psi }}}{\partial x^{c}\,\partial x^{a}}},{\frac {\partial {\vec {\Psi }}}{\partial x^{b}}}\right\rangle +\left\langle {\frac {\partial {\vec {\Psi }}}{\partial x^{a}}},{\frac {\partial ^{2}{\vec {\Psi }}}{\partial x^{c}\,\partial x^{b}}}\right\rangle }$

implies for a basis ${\displaystyle x^{i},x^{j},x^{k}}$, using the symmetry of the scalar product and swapping the order of partial differentiations:

${\displaystyle {\begin{pmatrix}{\frac {\partial g_{jk}}{\partial x^{i}}}\\{\frac {\partial g_{ki}}{\partial x^{j}}}\\{\frac {\partial g_{ij}}{\partial x^{k}}}\end{pmatrix}}={\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}}{\begin{pmatrix}\left\langle {\frac {\partial {\vec {\Psi }}}{\partial x^{i}}},{\frac {\partial ^{2}{\vec {\Psi }}}{\partial x^{j}\,\partial x^{k}}}\right\rangle \\\left\langle {\frac {\partial {\vec {\Psi }}}{\partial x^{j}}},{\frac {\partial ^{2}{\vec {\Psi }}}{\partial x^{k}\,\partial x^{i}}}\right\rangle \\\left\langle {\frac {\partial {\vec {\Psi }}}{\partial x^{k}}},{\frac {\partial ^{2}{\vec {\Psi }}}{\partial x^{i}\,\partial x^{j}}}\right\rangle \end{pmatrix}}}$

adding first row to second and subtracting third one:

${\displaystyle {\frac {\partial g_{jk}}{\partial x^{i}}}+{\frac {\partial g_{ki}}{\partial x^{j}}}-{\frac {\partial g_{ij}}{\partial x^{k}}}=2\left\langle {\frac {\partial {\vec {\Psi }}}{\partial x^{k}}},{\frac {\partial ^{2}{\vec {\Psi }}}{\partial x^{i}\,\partial x^{j}}}\right\rangle }$

and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric:

${\displaystyle g_{kl}{\Gamma ^{k}}_{ij}={\frac {1}{2}}\left({\frac {\partial g_{jl}}{\partial x^{i}}}+{\frac {\partial g_{li}}{\partial x^{j}}}-{\frac {\partial g_{ij}}{\partial x^{l}}}\right).}$

For a very simple example that captures the essence of the description above, draw a circle on a flat sheet of paper. Travel around the circle at a constant speed. The derivative of your velocity, your acceleration vector, always points radially inward. Roll this sheet of paper into a cylinder. Now the (Euclidean) derivative of your velocity has a component that sometimes points inward toward the axis of the cylinder depending on whether you're near a solstice or an equinox. (At the point of the circle when you are moving parallel to the axis, there is no inward acceleration. Conversely, at a point (1/4 of a circle later) when the velocity is along the cylinder's bend, the inward acceleration is maximum.) This is the (Euclidean) normal component. The covariant derivative component is the component parallel to the cylinder's surface, and is the same as that before you rolled the sheet into a cylinder.

Formal definition

A covariant derivative is a (Koszul) connection on the tangent bundle and other tensor bundles: it differentiates vector fields in a way analogous to the usual differential on functions. The definition extends to a differentiation on the duals of vector fields (i.e. covector fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction).

Functions

Given a point ${\displaystyle p\in M}$ of the manifold ${\displaystyle M}$, a real function ${\displaystyle f:M\rightarrow \mathbb {R} }$ on the manifold and a tangent vector ${\displaystyle \mathbf {v} \in T_{p}M}$, the covariant derivative of f at p along v is the scalar at p, denoted ${\displaystyle \left(\nabla _{\mathbf {v} }f\right)_{p}}$, that represents the principal part of the change in the value of f when the argument of f is changed by the infinitesimal displacement vector v. (This is the differential of f evaluated against the vector v.) Formally, there is a differentiable curve ${\displaystyle \phi :[-1,1]\to M}$ such that ${\displaystyle \phi (0)=p}$ and ${\displaystyle \phi '(0)=\mathbf {v} }$, and the covariant derivative of f at p is defined by

${\displaystyle \left(\nabla _{\mathbf {v} }f\right)_{p}=\left(f\circ \phi \right)'\left(0\right)=\lim _{t\to 0}{\frac {f\left[\phi \left(t\right)\right]-f\left[p\right]}{t}}.}$

When ${\displaystyle \mathbf {v} :M\rightarrow T_{p}M}$ is a vector field on ${\displaystyle M}$, the covariant derivative ${\displaystyle \nabla _{\mathbf {v} }f:M\rightarrow \mathbb {R} }$ is the function that associates with each point p in the common domain of f and v the scalar ${\displaystyle \left(\nabla _{\mathbf {v} }f\right)_{p}}$. This coincides with the usual Lie derivative of f along the vector field v.

Vector fields

Given a point ${\displaystyle p}$ of the manifold ${\displaystyle M}$, a vector field ${\displaystyle \mathbf {u} :M\rightarrow T_{p}M}$ defined in a neighborhood of p and a tangent vector ${\displaystyle \mathbf {v} \in T_{p}M}$, the covariant derivative of u at p along v is the tangent vector at p, denoted ${\displaystyle (\nabla _{\mathbf {v} }\mathbf {u} )_{p}}$, such that the following properties hold (for any tangent vectors v, x and y at p, vector fields u and w defined in a neighborhood of p, scalar values g and h at p, and scalar function f defined in a neighborhood of p):

1. ${\displaystyle \left(\nabla _{\mathbf {v} }\mathbf {u} \right)_{p}}$ is linear in ${\displaystyle \mathbf {v} }$ so

   ${\displaystyle \left(\nabla _{g\mathbf {x} +h\mathbf {y} }\mathbf {u} \right)_{p}=\left(\nabla _{\mathbf {x} }\mathbf {u} \right)_{p}g+\left(\nabla _{\mathbf {y} }\mathbf {u} \right)_{p}h}$

2. ${\displaystyle \left(\nabla _{\mathbf {v} }\mathbf {u} \right)_{p}}$ is additive in ${\displaystyle \mathbf {u} }$ so:

   ${\displaystyle \left(\nabla _{\mathbf {v} }\left[\mathbf {u} +\mathbf {w} \right]\right)_{p}=\left(\nabla _{\mathbf {v} }\mathbf {u} \right)_{p}+\left(\nabla _{\mathbf {v} }\mathbf {w} \right)_{p}}$

3. ${\displaystyle (\nabla _{\mathbf {v} }\mathbf {u} )_{p}}$ obeys the product rule; i.e., where ${\displaystyle \nabla _{\mathbf {v} }f}$ is defined above,

   ${\displaystyle \left(\nabla _{\mathbf {v} }\left[f\mathbf {u} \right]\right)_{p}=f(p)\left(\nabla _{\mathbf {v} }\mathbf {u} )_{p}+(\nabla _{\mathbf {v} }f\right)_{p}\mathbf {u} _{p}}$.

Note that ${\displaystyle \left(\nabla _{\mathbf {v} }\mathbf {u} \right)_{p}}$ depends not only on the value of u at p but also on values of u in an infinitesimal neighbourhood of p because of the last property, the product rule.

If u and v are both vector fields defined over a common domain, then ${\displaystyle \nabla _{\mathbf {v} }\mathbf {u} }$ denotes the vector field whose value at each point p of the domain is the tangent vector ${\displaystyle \left(\nabla _{\mathbf {v} }\mathbf {u} \right)_{p}}$.

Covector fields

Given a field of covectors (or one-form) ${\displaystyle \alpha }$ defined in a neighborhood of p, its covariant derivative ${\displaystyle (\nabla _{\mathbf {v} }\alpha )_{p}}$ is defined in a way to make the resulting operation compatible with tensor contraction and the product rule. That is, ${\displaystyle (\nabla _{\mathbf {v} }\alpha )_{p}}$ is defined as the unique one-form at p such that the following identity is satisfied for all vector fields u in a neighborhood of p

${\displaystyle \left(\nabla _{\mathbf {v} }\alpha \right)_{p}\left(\mathbf {u} _{p}\right)=\nabla _{\mathbf {v} }\left[\alpha \left(\mathbf {u} \right)\right]_{p}-\alpha _{p}\left[\left(\nabla _{\mathbf {v} }\mathbf {u} \right)_{p}\right].}$

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 by imposing the following identities for every pair of tensor fields ${\displaystyle \varphi }$ and ${\displaystyle \psi }$ in a neighborhood of the point p:

${\displaystyle \nabla _{\mathbf {v} }\left(\varphi \otimes \psi \right)_{p}=\left(\nabla _{\mathbf {v} }\varphi \right)_{p}\otimes \psi (p)+\varphi (p)\otimes \left(\nabla _{\mathbf {v} }\psi \right)_{p},}$

and for ${\displaystyle \varphi }$ and ${\displaystyle \psi }$ of the same valence

${\displaystyle \nabla _{\mathbf {v} }(\varphi +\psi )_{p}=(\nabla _{\mathbf {v} }\varphi )_{p}+(\nabla _{\mathbf {v} }\psi )_{p}.}$

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 (p, q). Consider T to be a differentiable multilinear map of smooth sections α¹, α², …, α^q of the cotangent bundle T^∗M and of sections X₁, X₂, …, X_p 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

${\displaystyle {\begin{aligned}(\nabla _{Y}T)\left(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots \right)=&{}Y\left(T\left(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots \right)\right)\\&{}-T\left(\nabla _{Y}\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots \right)-T\left(\alpha _{1},\nabla _{Y}\alpha _{2},\ldots ,X_{1},X_{2},\ldots \right)-\cdots \\&{}-T\left(\alpha _{1},\alpha _{2},\ldots ,\nabla _{Y}X_{1},X_{2},\ldots \right)-T\left(\alpha _{1},\alpha _{2},\ldots ,X_{1},\nabla _{Y}X_{2},\ldots \right)-\cdots \end{aligned}}}$

Coordinate description

This section uses the Einstein summation convention.

Given coordinate functions

${\displaystyle x^{i},\ i=0,1,2,\dots ,}$

any tangent vector can be described by its components in the basis

${\displaystyle \mathbf {e} _{i}={\frac {\partial }{\partial x^{i}}}.}$

The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination ${\displaystyle \Gamma ^{k}\mathbf {e} _{k}}$. To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field ${\displaystyle \mathbf {e} _{i}}$ along ${\displaystyle \mathbf {e} _{j}}$.

${\displaystyle \nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}={\Gamma ^{k}}_{ij}\mathbf {e} _{k},}$

the coefficients ${\displaystyle \Gamma _{ij}^{k}}$ are the components of the connection with respect to a system of local coordinates. In the theory of Riemannian and pseudo-Riemannian manifolds, the components of the Levi-Civita connection with respect to a system of local coordinates are called Christoffel symbols.

Then using the rules in the definition, we find that for general vector fields ${\displaystyle \mathbf {v} =v^{j}\mathbf {e} _{j}}$ and ${\displaystyle \mathbf {u} =u^{i}\mathbf {e} _{i}}$ we get

${\displaystyle {\begin{aligned}\nabla _{\mathbf {v} }\mathbf {u} &=\nabla _{v^{j}\mathbf {e} _{j}}u^{i}\mathbf {e} _{i}\\&=v^{j}\nabla _{\mathbf {e} _{j}}u^{i}\mathbf {e} _{i}\\&=v^{j}u^{i}\nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}+v^{j}\mathbf {e} _{i}\nabla _{\mathbf {e} _{j}}u^{i}\\&=v^{j}u^{i}{\Gamma ^{k}}_{ij}\mathbf {e} _{k}+v^{j}{\partial u^{i} \over \partial x^{j}}\mathbf {e} _{i}\end{aligned}}}$

so

${\displaystyle \nabla _{\mathbf {v} }\mathbf {u} =\left(v^{j}u^{i}{\Gamma ^{k}}_{ij}+v^{j}{\partial u^{k} \over \partial x^{j}}\right)\mathbf {e} _{k}.}$

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

${\displaystyle \nabla _{\mathbf {e} _{j}}\mathbf {u} =\nabla _{j}\mathbf {u} =\left({\frac {\partial u^{i}}{\partial x^{j}}}+u^{k}{\Gamma ^{i}}_{kj}\right)\mathbf {e} _{i}}$

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

For covectors similarly we have

${\displaystyle \nabla _{\mathbf {e} _{j}}{\mathbf {\theta } }=\left({\frac {\partial \theta _{i}}{\partial x^{j}}}-\theta _{k}{\Gamma ^{k}}_{ij}\right){\mathbf {e} ^{*}}^{i}}$

where ${\displaystyle {\mathbf {e} ^{*}}^{i}(\mathbf {e} _{j})={\delta ^{i}}_{j}}$.

The covariant derivative of a type (r, s) tensor field along ${\displaystyle e_{c}}$ is given by the expression:

${\displaystyle {\begin{aligned}{(\nabla _{e_{c}}T)^{a_{1}\ldots a_{r}}}_{b_{1}\ldots b_{s}}={}&{\frac {\partial }{\partial x^{c}}}{T^{a_{1}\ldots a_{r}}}_{b_{1}\ldots b_{s}}\\&+\,{\Gamma ^{a_{1}}}_{dc}{T^{da_{2}\ldots a_{r}}}_{b_{1}\ldots b_{s}}+\cdots +{\Gamma ^{a_{r}}}_{dc}{T^{a_{1}\ldots a_{r-1}d}}_{b_{1}\ldots b_{s}}\\&-\,{\Gamma ^{d}}_{b_{1}c}{T^{a_{1}\ldots a_{r}}}_{db_{2}\ldots b_{s}}-\cdots -{\Gamma ^{d}}_{b_{s}c}{T^{a_{1}\ldots a_{r}}}_{b_{1}\ldots b_{s-1}d}.\end{aligned}}}$

Or, in words: take the partial derivative of the tensor and add: ${\displaystyle +{\Gamma ^{a_{i}}}_{dc}}$ for every upper index ${\displaystyle a_{i}}$, and ${\displaystyle -{\Gamma ^{d}}_{b_{i}c}}$ for every lower index ${\displaystyle b_{i}}$.

If instead of a tensor, one is trying to differentiate a tensor density (of weight +1), then you also add a term

${\displaystyle -{\Gamma ^{d}}_{dc}{T^{a_{1}\ldots a_{r}}}_{b_{1}\ldots b_{s}}.}$

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

${\displaystyle \left({\sqrt {-g}}\right)_{;c}=\left({\sqrt {-g}}\right)_{,c}-{\sqrt {-g}}\,{\Gamma ^{d}}_{dc}}$

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.

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:

${\displaystyle \nabla _{e_{j}}\mathbf {v} \ {\stackrel {\mathrm {def} }{=}}\ {v^{s}}_{;j}\mathbf {e} _{s}\;\;\;\;\;\;{v^{i}}_{;j}={v^{i}}_{,j}+v^{k}{\Gamma ^{i}}_{kj}}$

In case two or more indexes after the semicolon, all them must be understood as covariant derivatives:

${\displaystyle \nabla _{e_{k}}\left(\nabla _{e_{j}}\mathbf {v} \right)\ {\stackrel {\mathrm {def} }{=}}\ {v^{s}}_{;kj}\mathbf {e} _{s}}$

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

${\displaystyle \nabla _{e_{j}}\mathbf {v} \ {\stackrel {\mathrm {def} }{=}}\ {v^{i}}_{||j}={v^{i}}_{|j}+v^{k}{\Gamma ^{i}}_{kj}}$

Covariant derivative by field type

For a scalar field ${\displaystyle \displaystyle \phi \,}$, covariant differentiation is simply partial differentiation:

${\displaystyle \displaystyle \phi _{;a}\equiv \partial _{a}\phi }$

For a contravariant vector field ${\displaystyle \lambda ^{a}\,}$, we have:

${\displaystyle {\lambda ^{a}}_{;b}\equiv \partial _{b}\lambda ^{a}+{\Gamma ^{a}}_{bc}\lambda ^{c}}$

For a covariant vector field ${\displaystyle \lambda _{a}\,}$, we have:

${\displaystyle \lambda _{a;c}\equiv \partial _{c}\lambda _{a}-{\Gamma ^{b}}_{ca}\lambda _{b}}$

For a type (2,0) tensor field ${\displaystyle \tau ^{ab}\,}$, we have:

${\displaystyle {\tau ^{ab}}_{;c}\equiv \partial _{c}\tau ^{ab}+{\Gamma ^{a}}_{cd}\tau ^{db}+{\Gamma ^{b}}_{cd}\tau ^{ad}}$

For a type (0,2) tensor field ${\displaystyle \tau _{ab}\,}$, we have:

${\displaystyle \tau _{ab;c}\equiv \partial _{c}\tau _{ab}-{\Gamma ^{d}}_{ca}\tau _{db}-{\Gamma ^{d}}_{cb}\tau _{ad}}$

For a type (1,1) tensor field ${\displaystyle {\tau ^{a}}_{b}\,}$, we have:

${\displaystyle {\tau ^{a}}_{b;c}\equiv \partial _{c}{\tau ^{a}}_{b}+{\Gamma ^{a}}_{cd}{\tau ^{d}}_{b}-{\Gamma ^{d}}_{cb}{\tau ^{a}}_{d}}$

The notation above is meant in the sense

${\displaystyle {\tau ^{ab}}_{;c}\equiv \left(\nabla _{\mathbf {e} _{c}}\tau \right)^{ab}}$

Properties

In general, covariant derivatives do not commute. By example, the covariant derivatives of vector field ${\displaystyle \lambda _{a;bc}\neq \lambda _{a;cb}\,}$. Riemann tensor ${\displaystyle {R^{d}}_{abc}\,}$is defined such that:

${\displaystyle \lambda _{a;bc}-\lambda _{a;cb}={R^{d}}_{abc}\lambda _{d}}$

or, equivalently,

${\displaystyle {\lambda ^{a}}_{;bc}-{\lambda ^{a}}_{;cb}=-{R^{a}}_{dbc}\lambda ^{d}}$

The covariant derivative of a (2,1)-tensor field fulfills:

${\displaystyle {\tau ^{ab}}_{;cd}-{\tau ^{ab}}_{;dc}=-{R^{a}}_{ecd}\tau ^{eb}-{R^{b}}_{ecd}\tau ^{ae}}$

The latter can be shown by taking (without loss of generality) that ${\displaystyle \tau ^{ab}=\lambda ^{a}\mu ^{b}\,}$.

Derivative along a curve

Since the covariant derivative ${\displaystyle \nabla _{X}T}$ of a tensor field ${\displaystyle T}$ at a point ${\displaystyle p}$ depends only on the value of the vector field ${\displaystyle X}$ at ${\displaystyle p}$ one can define the covariant derivative along a smooth curve ${\displaystyle \gamma (t)}$ in a manifold:

${\displaystyle D_{t}T=\nabla _{{\dot {\gamma }}(t)}T.}$

Note that the tensor field ${\displaystyle T}$ only needs to be defined on the curve ${\displaystyle \gamma (t)}$ for this definition to make sense.

In particular, ${\displaystyle {\dot {\gamma }}(t)}$ is a vector field along the curve ${\displaystyle \gamma }$ itself. If ${\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)}$ vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a positive-definite 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 that allows vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system.

There is however another generalization of directional derivatives which is canonical: the Lie derivative, which 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, not merely at a single point. 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.

Note that the antisymmetrized covariant derivative ∇_uv − ∇_vu, and the Lie derivative L_uv differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization is the Lie derivative.

See also

• Affine connection
• Christoffel symbols
• Connection (algebraic framework)
• Connection (mathematics)
• Connection (vector bundle)
• Connection form
• Exterior covariant derivative
• Gauge covariant derivative
• Introduction to the mathematics of general relativity
• Levi-Civita connection
• Parallel transport
• Ricci calculus
• Tensor derivative (continuum mechanics)
• List of formulas in Riemannian geometry

Notes

1. Einstein, Albert (1922). "The General Theory of Relativity". The Meaning of Relativity.
2. Ricci, G.; Levi-Civita, T. (1901). "Méthodes de calcul différential absolu et leurs applications". Mathematische Annalen. 54: 125–201. doi:10.1007/bf01454201.
3. Riemann, G. F. B. (1866). "Über die Hypothesen, welche der Geometrie zu Grunde liegen". Gesammelte Mathematische Werke.; reprint, ed. Weber, H. (1953), New York: Dover.
4. Christoffel, E. B. (1869). "Über die Transformation der homogenen Differentialausdrücke zweiten Grades". Journal für die reine und angewandte Mathematik. 70: 46–70.
5. cf. with Cartan, É (1923). "Sur les variétés à connexion affine et la theorie de la relativité généralisée". Annales, École Normale. 40: 325–412.
6. Koszul, J. L. (1950). "Homologie et cohomologie des algebres de Lie". Bulletin de la Société Mathématique. 78: 65–127.
7. The covariant derivative is also denoted variously by ${\displaystyle \partial }$_vu, D_vu, or other notations.
8. 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.

References

• Kobayashi, Shoshichi; Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley Interscience. ISBN 0-471-15733-3.
• I.Kh. Sabitov (2001) [1994], "Covariant differentiation", Encyclopedia of Mathematics, EMS Press
• Sternberg, Shlomo (1964). Lectures on Differential Geometry. Prentice-Hall.
• Spivak, Michael (1999). A Comprehensive Introduction to Differential Geometry (Volume Two). Publish or Perish, Inc.