View source for 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 [[#References|[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|connection]] and [[Parallel displacement(2)|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c0268701.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c0268702.png" /> as well as the parallel displacement of vectors and, more generally, tensors, associated with it. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c0268703.png" /> be a smooth vector field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c0268704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c0268705.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c0268706.png" /> be a tensor field of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c0268707.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c0268708.png" /> times contravariant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c0268709.png" /> times covariant; by the covariant derivative (with respect to the given connection) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687010.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687011.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687012.png" /> one means the tensor (of the same type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687013.png" />)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687015.png" /> is the point on the integral curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687016.png" /> of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687017.png" /> with initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687020.png" /> are, respectively, the localizations (values) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687021.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687024.png" /> is the result of the parallel displacement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687025.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687026.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687027.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687028.png" />. Thus the basic idea behind the definition of the covariant derivative of a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687029.png" /> along a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687030.png" /> is that, in view of the absence of a natural relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687032.png" />, as they belong to different fibres of the tensor bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687033.png" />, that is, they are in tensor spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687034.png" /> over different tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687037.png" />, the difference between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687038.png" /> and the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687039.png" /> under the parallel displacement along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687040.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687041.png" /> serves as the  "increment"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687042.png" />; one then takes the limit of the ratio of this  "increment"  to the increment of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687043.png" /> in the usual way. If, in particular, for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687044.png" /> near to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687045.png" /> the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687046.png" /> is obtained by parallel displacement of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687047.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687049.png" />, and therefore, in general, the covariant derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687050.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687051.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687052.png" /> defines the initial rate of the difference of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687053.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687054.png" /> from the result of the parallel displacement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687055.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687056.png" />. For tensor fields of zero valency, that is, for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687057.png" /> in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687058.png" /> of differentiable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687059.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687060.png" /></td> </tr></table>

which leads to the identification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687061.png" /> with the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687062.png" /> along the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687063.png" />, that is, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687064.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687065.png" /> one has, by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687066.png" /> for any tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687067.png" />.

The introduction of a covariant derivative enables one to define the covariant differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687068.png" /> of a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687069.png" /> along a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687070.png" /> as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687071.png" /></td> </tr></table>

which can be regarded as the principal linear part of the  "increment"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687072.png" /> (in the sense described above) under the displacement along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687073.png" /> of the point by an infinitesimal segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687074.png" />.

The knowledge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687075.png" /> for a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687076.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687077.png" /> at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687078.png" /> along each vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687079.png" /> enables one to introduce for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687080.png" />: 1) the covariant differential field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687081.png" /> as a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687082.png" />-form with values in the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687083.png" />, defined on the vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687084.png" /> by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687085.png" />; 2) the covariant derivative field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687086.png" /> as a tensor field of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687087.png" />, corresponding canonically to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687088.png" /> and acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687089.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687090.png" /> and vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687091.png" /> according to the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687092.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687093.png" /></td> </tr></table>

By the covariant differential one usually means not the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687094.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687095.png" /> itself but its values at the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687096.png" />, and in this interpretation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687097.png" /> is also converted into a tensor field of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687098.png" /> the localization of which, in particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c02687099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870100.png" />, is the same as the covariant differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870101.png" /> along the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870102.png" />, introduced above. The covariant derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870103.png" /> is sometimes called the gradient of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870104.png" /> and the derivative, the covariant differential.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870105.png" /> are local coordinates, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870106.png" /> denotes the corresponding basis of the space of vector fields, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870107.png" /> denotes the dual basis for the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870108.png" />-forms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870110.png" /> are the coordinates of vector and tensor fields in these bases, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870111.png" /> are the coefficients of an affine connection introduced on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870112.png" /> (cf. [[Linear connection|Linear connection]]), then, denoting by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870113.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870114.png" /> the components of the tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870115.png" />, one obtains the following expressions (as an example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870117.png" /> have been chosen):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870118.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870119.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870120.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870121.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870122.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870123.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870124.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870125.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870126.png" /> is the operation of contraction (cf. [[Contraction of a tensor|Contraction of a tensor]]) with respect to the third contravariant and second covariant indices.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870127.png" /> is an affine space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870128.png" /> are affine coordinates, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870129.png" /> is the ordinary derivative of the tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870130.png" /> along the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870131.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870132.png" /> are the partial derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870133.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870134.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870135.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870136.png" /> is the ordinary differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870137.png" /> along the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870138.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870139.png" /> along a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870140.png" /> is given by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870141.png" />, the equation of a geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870142.png" /> is written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870143.png" />, the integrability condition for a system of equations in covariant derivatives of the first order is reduced to an equation for the alternating difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870144.png" />; 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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870145.png" />. Generally speaking, the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870146.png" /> 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|curvature tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870147.png" /> and [[Torsion tensor|torsion tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870148.png" />, which together characterize the difference between the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870149.png" /> and affine space. For example,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870150.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870151.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870152.png" /></td> </tr></table>

(the Ricci identity);

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870153.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870154.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870155.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870156.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870157.png" /> is the commutator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870158.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870159.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870160.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870161.png" /></td> </tr></table>

The definition of covariant differentiation remains valid in the more general case when instead of a cross section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870162.png" /> of the tensor bundle with an affine connection one considers a cross section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870163.png" /> of an arbitrary (real or complex) [[Vector bundle|vector bundle]] associated with some [[Principal fibre bundle|principal fibre bundle]] with [[Connection|connection]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870164.png" /> and with a structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870165.png" /> 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 [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Ricci,   T. Levi-Civita,   "Méthodes de calcul différentiel absolu et leurs applications"  ''Math. Ann.'' , '''54'''  (1901)  pp. 125–201</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.P. Norden,   "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Lichnerowicz,   "Global theory of connections and holonomy groups" , Noordhoff  (1955)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Helgason,   "Differential geometry and symmetric spaces" , Acad. Press  (1962)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R.L. Bishop,   R.J. Crittenden,   "Geometry of manifolds" , Acad. Press  (1964)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D. Gromoll,   W. Klingenberg,   W. Meyer,   "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1–2''' , Interscience  (1963–1969)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  R. Sulanke,   P. Wintgen,   "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft.  (1972)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  (1970–1975)  pp. 1–5</TD></TR></table>



====Comments====
The phrase  "covariant differential"  is more commonly used for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870166.png" /> tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870167.png" /> (instead of  "covariant derivative"  as in the article above); that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870168.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026870/c026870169.png" /> are more or less identified.