absolute differentiation
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 [1]). 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 [9] 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.
References
| [1] | G. Ricci, T. Levi-Civita, "Méthodes de calcul différentiel absolu et leurs applications" Math. Ann. , 54 (1901) pp. 125–201 |
| [2] | P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) |
| [3] | A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian) |
| [4] | A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1955) (Translated from French) |
| [5] | S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) |
| [6] | R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964) |
| [7] | D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) |
| [8] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1969) |
| [9] | 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 |
| [10] | R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft. (1972) |
| [11] | 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.
