# 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

### Connection (algebraic framework)

the covariant differential of a

### 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.

### Tensor derivative (continuum mechanics)

**curl of a tensorDerivativesderivatives of the invariants**

### Mathematics

**mathematicalmathmathematician**

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