An operation that defines in an invariant way the notions of a derivative and a differential for fields of geometric objects on manifolds, such as vectors, tensors, forms, etc. The basic concepts of the theory of covariant differentiation were given (under the name of absolute differential calculus) at the end of the 19th century in papers by G. Ricci, and in its most complete form in 1901 by him in collaboration with T. Levi-Civita (see ). At first, the theory of covariant differentiation was constructed on Riemannian manifolds and was intended in the first instance for the investigation of the invariants of differential forms. The definition and properties of covariant differentiation subsequently proved to be related in a natural way with the notions of connection and parallel displacement on manifolds, which were introduced later. Nowadays the theory of covariant differentiation is developed within the general framework of the theory of connections. As a device of tensor analysis, covariant differentiation is widely used in theoretical physics, particularly in the general theory of relativity.
Let an affine connection be given on an -dimensional manifold as well as the parallel displacement of vectors and, more generally, tensors, associated with it. Let be a smooth vector field, , , and let be a tensor field of type , that is, times contravariant and times covariant; by the covariant derivative (with respect to the given connection) of at along one means the tensor (of the same type )
where is the point on the integral curve of the vector field with initial condition , and are, respectively, the localizations (values) of at and , and is the result of the parallel displacement of along from to . Thus the basic idea behind the definition of the covariant derivative of a tensor field along a vector field is that, in view of the absence of a natural relation between and , as they belong to different fibres of the tensor bundle over , that is, they are in tensor spaces over different tangent spaces and to , the difference between and the image of under the parallel displacement along to serves as the "increment" of ; one then takes the limit of the ratio of this "increment" to the increment of the argument in the usual way. If, in particular, for points near to the field is obtained by parallel displacement of the tensor along , then , and therefore, in general, the covariant derivative of at along defines the initial rate of the difference of along from the result of the parallel displacement of along . For tensor fields of zero valency, that is, for functions in the ring of differentiable functions on ,
which leads to the identification of with the derivative of along the vector , that is, with . When one has, by definition, for any tensor field .
The introduction of a covariant derivative enables one to define the covariant differential of a tensor field along a smooth curve as
which can be regarded as the principal linear part of the "increment" of (in the sense described above) under the displacement along of the point by an infinitesimal segment .
The knowledge of for a tensor field of type at each point along each vector field enables one to introduce for : 1) the covariant differential field as a tensor -form with values in the module , defined on the vectors of by the formula ; 2) the covariant derivative field as a tensor field of type , corresponding canonically to the form and acting on -forms and vectors according to the formula
By the covariant differential one usually means not the -form itself but its values at the vectors , and in this interpretation is also converted into a tensor field of type the localization of which, in particular, when and , is the same as the covariant differential along the curve , introduced above. The covariant derivative is sometimes called the gradient of the tensor and the derivative, the covariant differential.
If are local coordinates, denotes the corresponding basis of the space of vector fields, denotes the dual basis for the space of -forms, and are the coordinates of vector and tensor fields in these bases, and are the coefficients of an affine connection introduced on the manifold (cf. Linear connection), then, denoting by or the components of the tensor field , one obtains the following expressions (as an example, , have been chosen):
where is the operation of contraction (cf. Contraction of a tensor) with respect to the third contravariant and second covariant indices.
If is an affine space and are affine coordinates, then is the ordinary derivative of the tensor field along the vector field , the are the partial derivatives of at with respect to , and is the ordinary differential of along the curve . Thus the covariant derivative emerges as a generalization of ordinary differentiation for which the well-known relationships between the first-order partial derivatives and differentials remain valid.
The value of covariant differentiation is that it provides a convenient analytic apparatus for the study and description of the properties of geometric objects and operations in invariant form. For example, the condition for parallel displacement of a tensor along a curve is given by the equation , the equation of a geodesic is written in the form , the integrability condition for a system of equations in covariant derivatives of the first order is reduced to an equation for the alternating difference ; exterior differentiation of forms on a manifold and in bundles over it can also be expressed in terms of covariant differentiation; and there are other examples.
The definition of higher covariant derivatives is given inductively: . Generally speaking, the tensor obtained in this way is not symmetric in the last covariant indices; higher covariant derivatives along different vector fields also depend on the order of differentiation. The alternating differences of the covariant derivatives of higher orders are expressed in terms of the curvature tensor and torsion tensor , which together characterize the difference between the manifold and affine space. For example,
(the Ricci identity);
where is the commutator of and , and
The definition of covariant differentiation remains valid in the more general case when instead of a cross section of the tensor bundle with an affine connection one considers a cross section of an arbitrary (real or complex) vector bundle associated with some principal fibre bundle with connection and with a structure group which acts on the fibre by means of a representation in the group of non-singular matrices. There exist definitions of covariant differentiation in the more general situation when the bundle is not necessarily a vector bundle. The common part of these definitions  consists in the analytic expression for the parallel displacement of an object or in the condition of being parallel of a cross section which is defined by the requirement that its covariant differential be zero. There are also similar approaches for infinite-dimensional manifolds.
|||G. Ricci, T. Levi-Civita, "Méthodes de calcul différentiel absolu et leurs applications" Math. Ann. , 54 (1901) pp. 125–201|
|||P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)|
|||A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)|
|||A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1955) (Translated from French)|
|||S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962)|
|||R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964)|
|||D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)|
|||S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1969)|
|||Ü. Lumiste, "Connection theory in bundle spaces" J. Soviet Math. , 1 (1973) pp. 363–390 Itogi Nauk. Algebra. Topol. Geom. 1969 (1971) pp. 123–168|
|||R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft. (1972)|
|||M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1970–1975) pp. 1–5|
The phrase "covariant differential" is more commonly used for the tensor field (instead of "covariant derivative" as in the article above); that is, and are more or less identified.
Covariant differentiation. I.Kh. Sabitov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Covariant_differentiation&oldid=13604