Covariant derivative

covariant differentiationtensor derivativecovariant differentialabsolute derivativeChristoffel symbolcovariantcovariant derivativesCovariant derivatives or differentialscovariant differential operatorcovariant'' divergence
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.wikipedia
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Connection (mathematics)

connectionconnectionsconnected
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.
An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction.

Affine connection

connectionaffineaffine connections
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.
This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle.

Connection (principal bundle)

principal connectionconnectionconnections
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 particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold.

Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection
Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) 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. In particular, Koszul connections eliminated the need for awkward manipulations of Christoffel symbols (and other analogous non-tensorial objects) in differential geometry.
In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric.

Directional derivative

normal derivativedirectionalderivative
Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) 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. 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 directional derivative onto the manifold's tangent space.
(see Exterior derivative), (see Covariant derivative), (see Lie derivative), or (see ), can be defined as follows.

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
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.
In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.

Differential geometry

differentialdifferential geometerdifferential geometry and topology
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.
The notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor.

Connection (vector bundle)

connectionKoszul connectionlinear connection
The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection.
A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero.

Elwin Bruno Christoffel

ChristoffelElwin ChristoffelChristoffel, Elwin Bruno
Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) 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.
In a famous 1869 paper on the equivalence problem for differential forms in n variables, published in Crelle's Journal, he introduced the fundamental technique later called covariant differentiation and used it to define the Riemann–Christoffel tensor (the most common method used to express the curvature of Riemannian manifolds).

Parallel transport

parallelparallel-transporttransported
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.
If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection.

General covariance

diffeomorphism invariancegenerally covariantcovariant
The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation.

Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) 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.
The parallel transport maps are related to the covariant derivative by

Gregorio Ricci-Curbastro

RicciGregorio Ricci CurbastroRicci-Curbastro
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.
In fact, it was essentially Christoffel’s idea of covariant differentiation that allowed Ricci-Curbastro to make the greatest progress.

Curvature of Riemannian manifolds

curvatureabstract definition of curvaturecurvature of a Riemannian manifold
The curvature of a Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connection (or covariant differentiation) \nabla and Lie bracket by the following formula:

Tensor

tensorsorderclassical treatment of tensors
In particular, Koszul connections eliminated the need for awkward manipulations of Christoffel symbols (and other analogous non-tensorial objects) in differential geometry.
Ricci calculus is the modern formalism and notation for tensor indices: indicating inner and outer products, covariance and contravariance, summations of tensor components, symmetry and antisymmetry, and partial and covariant derivatives.

Gauge covariant derivative

covariant derivative(gauge) covariant derivative
The gauge covariant derivative is a variation of the covariant derivative used in general relativity.

Tensor field

tensor analysistensortensor bundle
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. The covariant derivative of a tensor field is presented as an extension of the same concept.
This property is used to check, for example, that even though the Lie derivative and covariant derivative are not tensors, the torsion and curvature tensors built from them are.

Torsion tensor

torsiontorsion-freetorsion form
Note that the antisymmetrized covariant derivative ∇ u v − ∇ v u, and the Lie derivative L u v differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization is the Lie derivative.
Let M be a manifold with an affine connection on the tangent bundle (aka covariant derivative) ∇.

List of formulas in Riemannian geometry

formulas in Riemannian geometryreturn to article
The covariant derivative of a vector field with components v^i is given by:

Lie derivative

Lie bracketLie commutatorcommuting vector fields
This coincides with the usual Lie derivative of f along the vector field v.
Alternatively, if we are using a torsion-free connection (e.g., the Levi Civita connection), then the partial derivative \partial_a can be replaced with the covariant derivative which means replacing with (by abuse of notation) where the are the Christoffel coefficients.

Tensor density

tensor densitiesdensityrelative tensor
If instead of a tensor, one is trying to differentiate a tensor density (of weight +1), then you also add a term
When using the metric connection (Levi-Civita connection), the covariant derivative of an even tensor density is defined as

Semicolon

;؛ Semicolon [;]
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 differential geometry, a semicolon preceding an index is used to indicate the covariant derivative of a function with respect to the coordinate associated with that index.

Mathematics

mathematicalmathmathematician
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.