Covariant differentiation

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 $ n $- dimensional manifold $ M $ as well as the parallel displacement of vectors and, more generally, tensors, associated with it. Let $ X $ be a smooth vector field, $ X _ {p} \neq 0 $, $ p \in M $, and let $ U $ be a tensor field of type $ ( r, s) $, that is, $ r $ times contravariant and $ s $ times covariant; by the covariant derivative (with respect to the given connection) of $ U $ at $ p \in M $ along $ X $ one means the tensor (of the same type $ ( r, s) $)

$$ ( \nabla _ {X} U) _ {p} = \ \lim\limits _ {t \rightarrow 0 } \ \frac{\tau _ {t} ^ {-} 1 ( U _ {x ( t) } ) - U _ {p} }{t} , $$

where $ x ( t) $ is the point on the integral curve $ \gamma _ {X} $ of the vector field $ X $ with initial condition $ x ( 0) = p $, $ U _ {p} $ and $ U _ {x ( t) } $ are, respectively, the localizations (values) of $ U $ at $ p $ and $ x ( t) $, and $ \tau _ {t} ^ {-} 1 ( U _ {x ( t) } ) $ is the result of the parallel displacement of $ U _ {x ( t) } $ along $ \gamma _ {X} $ from $ x ( t) $ to $ p $. Thus the basic idea behind the definition of the covariant derivative of a tensor field $ U $ along a vector field $ X $ is that, in view of the absence of a natural relation between $ U _ {p} $ and $ U _ {x ( t) } $, as they belong to different fibres of the tensor bundle over $ M $, that is, they are in tensor spaces $ T _ {s} ^ {r} $ over different tangent spaces $ T _ {p} M $ and $ T _ {x ( t) } M $ to $ M $, the difference between $ U _ {p} \in T _ {s} ^ {r} ( T _ {p} M) $ and the image of $ U _ {x ( t) } \in T _ {s} ^ {r} ( T _ {x ( t) } M) $ under the parallel displacement along $ \gamma _ {X} $ to $ T _ {s} ^ {r} ( T _ {p} M) $ serves as the "increment" of $ U $; one then takes the limit of the ratio of this "increment" to the increment of the argument $ t $ in the usual way. If, in particular, for points $ x ( t) $ near to $ p $ the field $ U $ is obtained by parallel displacement of the tensor $ U _ {p} $ along $ \gamma _ {X } $, then $ ( \nabla _ {X} U) _ {p} = 0 $, and therefore, in general, the covariant derivative of $ U $ at $ p $ along $ X $ defines the initial rate of the difference of $ U $ along $ \gamma _ {X} $ from the result of the parallel displacement of $ U _ {p} $ along $ \gamma _ {X} $. For tensor fields of zero valency, that is, for functions $ f $ in the ring $ {\mathcal T} $ of differentiable functions on $ M $,

$$ ( \nabla _ {X} f) _ {p} = \ \lim\limits _ {t \rightarrow 0 } \ \frac{f ( x ( t)) - f ( p) }{t} , $$

which leads to the identification of $ ( \nabla _ {X} f) _ {p} $ with the derivative of $ f $ along the vector $ X _ {p} $, that is, with $ X _ {p} f $. When $ X _ {p} = 0 $ one has, by definition, $ ( \nabla _ {X} U) _ {p} = 0 $ for any tensor field $ U $.

The introduction of a covariant derivative enables one to define the covariant differential $ DU $ of a tensor field $ U $ along a smooth curve $ \gamma ( t) $ as

$$ ( DU) _ {\gamma ( 0) } = \ \left ( \nabla _ {\dot \gamma ( t) } U \right ) _ {\gamma ( 0) } dt, $$

which can be regarded as the principal linear part of the "increment" of $ U $( in the sense described above) under the displacement along $ \gamma $ of the point by an infinitesimal segment $ d \gamma = \dot \gamma ( 0) dt $.

The knowledge of $ \nabla _ {X} U $ for a tensor field $ U $ of type $ ( r, s) $ at each point $ p \in M $ along each vector field $ X $ enables one to introduce for $ U $: 1) the covariant differential field $ DU $ as a tensor $ 1 $- form with values in the module $ T _ {s} ^ {r} ( M) $, defined on the vectors of $ X $ by the formula $ ( DU) ( X) = \nabla _ {X} U $; 2) the covariant derivative field $ \nabla U $ as a tensor field of type $ ( r, s + 1) $, corresponding canonically to the form $ DU $ and acting on $ 1 $- forms $ \omega ^ {i} $ and vectors $ X _ {i} $ according to the formula

$$ ( \nabla U) ( \omega ^ {1} \dots \omega ^ {r} ; \ X _ {1} \dots X _ {s} , X) = $$

$$ = \ ( \nabla _ {X} U) ( \omega ^ {1} \dots \omega ^ {r} ; X _ {1} \dots X _ {s} ). $$

By the covariant differential one usually means not the $ 1 $- form $ DU $ itself but its values at the vectors $ X $, and in this interpretation $ ( DU) X $ is also converted into a tensor field of type $ ( r, s) $ the localization of which, in particular, when $ p = \gamma ( 0) $ and $ X = \dot \gamma $, is the same as the covariant differential $ ( DU) _ {\gamma ( 0) } $ along the curve $ \gamma ( t) $, introduced above. The covariant derivative $ \nabla U $ is sometimes called the gradient of the tensor $ U $ and the derivative, the covariant differential.

If $ x ^ {i} $ are local coordinates, $ e _ {i} = \partial / \partial x ^ {i} \mid _ {p} $ denotes the corresponding basis of the space of vector fields, $ e ^ {i} $ denotes the dual basis for the space of $ 1 $- forms, $ X ^ {i} $ and $ U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } $ are the coordinates of vector and tensor fields in these bases, and $ \Gamma _ {ij} $ are the coefficients of an affine connection introduced on the manifold $ M $( cf. Linear connection), then, denoting by $ \nabla _ {k} U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } $ or $ U _ {j _ {1} \dots j _ {s} , k } ^ {i _ {1} \dots i _ {r} } $ the components of the tensor field $ \nabla U $, one obtains the following expressions (as an example, $ r = 2 $, $ s = 1 $ have been chosen):

$$ ( \nabla _ {X} U) _ {j} ^ {i _ {1} i _ {2} } = $$

$$ = \ \left ( \frac{\partial U _ {j} ^ {i _ {1} i _ {2} } }{\partial x _ {k} } + \Gamma _ {km} ^ {i _ {1} } U _ {j} ^ {mi _ {2} } + \Gamma _ {km} ^ {i _ {2} } U _ {j} ^ {i _ {1} m } - \Gamma _ {kj} ^ {m} U _ {m} ^ {i _ {1} i _ {2} } \right ) X ^ {k\ } \equiv $$

$$ \equiv \ \nabla _ {k} U _ {j} ^ {i _ {1} i _ {2} } X ^ {k} ; $$

$$ ( DU _ {\gamma ( t) } ) _ {j} ^ {i _ {1} i _ {2} } = \nabla _ {k} U _ {j} ^ {i _ {1} i _ {2} } dx ^ {k} , $$

$$ dx ^ {k} = \dot{x} ^ {k} dt,\ \gamma ( t) = \{ x ^ {k} ( t) \} , $$

$$ DU = ( U _ {j,k} ^ {i _ {1} i _ {2} } e _ {i _ {1} } \otimes e _ {i _ {2} } \otimes e ^ {j} ) e ^ {k} , $$

$$ \nabla U = U _ {j,k} ^ {i _ {1} i _ {2} } e _ {i _ {1} } \otimes e _ {i _ {2} } \otimes e ^ {j} \otimes e ^ {k} , $$

$$ \nabla _ {X} U = ( DU) ( X) = U _ {j,k} ^ {i _ {1} i _ {2} } X ^ {k} e _ {i _ {1} } \otimes e _ {i _ {2} } \otimes e ^ {j} = C _ {2} ^ { 3 } ( \nabla U \otimes X), $$

where $ C _ {2} ^ { 3 } $ is the operation of contraction (cf. Contraction of a tensor) with respect to the third contravariant and second covariant indices.

If $ M $ is an affine space and $ x ^ {i} $ are affine coordinates, then $ ( \nabla _ {X} U) _ {p} $ is the ordinary derivative of the tensor field $ U $ along the vector field $ X $, the $ ( \nabla _ {k} U) _ {p} $ are the partial derivatives of $ U $ at $ p $ with respect to $ x ^ {k} $, and $ ( DU) _ {\gamma ( t) } $ is the ordinary differential of $ U $ along the curve $ \gamma ( t) $. 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 $ U $ along a curve $ \gamma $ is given by the equation $ \nabla _ {\dot \gamma } U = 0 $, the equation of a geodesic $ \gamma $ is written in the form $ \nabla _ {\dot \gamma } \dot \gamma $, the integrability condition for a system of equations in covariant derivatives of the first order is reduced to an equation for the alternating difference $ \nabla _ \lambda \nabla _ \mu U - \nabla _ \mu \nabla _ \lambda U $; 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: $ \nabla ^ {m} U = \nabla ( \nabla ^ {m - 1 } U) $. Generally speaking, the tensor $ \nabla ^ {m} U $ 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 $ R _ {jkl} ^ { i } $ and torsion tensor $ S _ {jk} ^ { i } $, which together characterize the difference between the manifold $ M $ and affine space. For example,

$$ \nabla _ \lambda \nabla _ \mu U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } - \nabla _ \mu \nabla _ \lambda U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } = $$

$$ = \ \sum _ {k = 1 } ^ { r } R _ {j \lambda \mu } ^ { i _ {k} } U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {k - 1 } ji _ {k + 1 } \dots i _ {r} } - $$

$$ - \sum _ {k = 1 } ^ { s } R _ {j _ {k} \lambda \mu } ^ { i } U _ {j _ {1} \dots j _ {k - 1 } ij _ {k + 1 } \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } - S _ {\lambda \mu } ^ { k } \nabla _ {k} U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } $$

(the Ricci identity);

$$ \nabla _ {X} \nabla _ {Y} U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } - \nabla _ {Y} \nabla _ {X} U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } = $$

$$ = \ \left ( \sum _ {k = 1 } ^ { r } R _ {j \lambda \mu } ^ { i _ {k} } U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {k - 1 } ji _ {k + 1 } \dots i _ {r} }\right . - $$

$$ - \left . \sum _ {k = 1 } ^ { s } R _ {j _ {k} \lambda \mu } ^ { i } U _ {j _ {1} \dots j _ {k - 1 } ij _ {k + 1 } \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } \right ) X ^ \lambda X ^ \mu + $$

$$ + \nabla _ {[ X, Y] } U _ {j _ {1} \dots j _ {s} } ^ {i _ {1} \dots i _ {r} } , $$

where $ [ X, Y] $ is the commutator of $ X $ and $ Y $, and

$$ R _ {jkl} ^ { i } = \ { \frac \partial {\partial x ^ {k} } } \Gamma _ {lj} ^ {i} - { \frac \partial {\partial x ^ {l} } } \Gamma _ {kj} ^ {i} + \Gamma _ {kp} ^ {i} \Gamma _ {lj} ^ {p} - \Gamma _ {kj} ^ {p} \Gamma _ {lp} ^ {i} , $$

$$ S _ {jk} ^ { i } = \Gamma _ {jk} ^ {i} - \Gamma _ {kj} ^ {i} . $$

The definition of covariant differentiation remains valid in the more general case when instead of a cross section $ U $ of the tensor bundle with an affine connection one considers a cross section $ \phi $ of an arbitrary (real or complex) vector bundle associated with some principal fibre bundle with connection $ \Gamma $ and with a structure group $ G $ 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

Comments

The phrase "covariant differential" is more commonly used for the $ ( r, s + 1) $ tensor field $ \nabla U $( instead of "covariant derivative" as in the article above); that is, $ DU $ and $ \nabla U $ are more or less identified.