Ricci calculus

Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.

Working with a main proponent of the exterior calculus Élie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:^[7]

In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.

Where a distinction is to be made between the space-like basis elements and a time-like element in the four-dimensional spacetime of classical physics, this is conventionally done through indices as follows:^[8]

• The lowercase Latin alphabet a, b, c, ... is used to indicate restriction to 3-dimensional Euclidean space, which take values 1, 2, 3 for the spatial components; and the time-like element, indicated by 0, is shown separately.
• The lowercase Greek alphabet α, β, γ, ... is used for 4-dimensional spacetime, which typically take values 0 for time components and 1, 2, 3 for the spatial components.

Some sources use 4 instead of 0 as the index value corresponding to time; in this article, 0 is used. Otherwise, in general mathematical contexts, any symbols can be used for the indices, generally running over all dimensions of the vector space.

The author(s) will usually make it clear whether a subscript is intended as an index or as a label.

For example, in 3-D Euclidean space and using Cartesian coordinates; the coordinate vector A = (A₁, A₂, A₃) = (A_x, A_y, A_z) shows a direct correspondence between the subscripts 1, 2, 3 and the labels x, y, z. In the expression A_i, i is interpreted as an index ranging over the values 1, 2, 3, while the x, y, z subscripts are only labels, not variables. In the context of spacetime, the index value 0 conventionally corresponds to the label t.

Indices themselves may be labelled using diacritic-like symbols, such as a hat (ˆ), bar (¯), tilde (˜), or prime (′) as in:

${\displaystyle X_{\hat {\phi }}\,,Y_{\bar {\lambda }}\,,Z_{\tilde {\eta }}\,,T_{\mu '}}$

to denote a possibly different basis for that index. An example is in Lorentz transformations from one frame of reference to another, where one frame could be unprimed and the other primed, as in:

${\displaystyle v^{\mu '}=v^{\nu }L_{\nu }{}^{\mu '}.}$

This is not to be confused with van der Waerden notation for spinors, which uses hats and overdots on indices to reflect the chirality of a spinor.

Ricci calculus, and index notation more generally, distinguishes between lower indices (subscripts) and upper indices (superscripts); the latter are not exponents, even though they may look as such to the reader only familiar with other parts of mathematics.

In the special case that the metric tensor is everywhere equal to the identity matrix, it is possible to drop the distinction between upper and lower indices, and then all indices could be written in the lower position. Coordinate formulae in linear algebra such as ${\displaystyle a_{ij}b_{jk}}$ for the product of matrices may be examples of this. But in general, the distinction between upper and lower indices should be maintained.

A lower index (subscript) indicates covariance of the components with respect to that index:

${\displaystyle A_{\alpha \beta \gamma \cdots }}$

An upper index (superscript) indicates contravariance of the components with respect to that index:

${\displaystyle A^{\alpha \beta \gamma \cdots }}$

A tensor may have both upper and lower indices:

${\displaystyle A_{\alpha }{}^{\beta }{}_{\gamma }{}^{\delta \cdots }.}$

Ordering of indices is significant, even when of differing variance. However, when it is understood that no indices will be raised or lowered while retaining the base symbol, covariant indices are sometimes placed below contravariant indices for notational convenience (e.g. with the generalized Kronecker delta).

The number of each upper and lower indices of a tensor gives its type: a tensor with p upper and q lower indices is said to be of type (p, q), or to be a type-(p, q) tensor.

The number of indices of a tensor, regardless of variance, is called the degree of the tensor (alternatively, its valence, order or rank, although rank is ambiguous). Thus, a tensor of type (p, q) has degree p + q.

The same symbol occurring twice (one upper and one lower) within a term indicates a pair of indices that are summed over:

${\displaystyle A_{\alpha }B^{\alpha }\equiv \sum _{\alpha }A_{\alpha }B^{\alpha }\quad {\text{or}}\quad A^{\alpha }B_{\alpha }\equiv \sum _{\alpha }A^{\alpha }B_{\alpha }\,.}$

The operation implied by such a summation is called tensor contraction:

${\displaystyle A_{\alpha }B^{\beta }\rightarrow A_{\alpha }B^{\alpha }\equiv \sum _{\alpha }A_{\alpha }B^{\alpha }\,.}$

This summation may occur more than once within a term with a distinct symbol per pair of indices, for example:

${\displaystyle A_{\alpha }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\equiv \sum _{\alpha }\sum _{\gamma }A_{\alpha }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }\,.}$

Other combinations of repeated indices within a term are considered to be ill-formed, such as

${\displaystyle A_{\alpha \alpha }{}^{\gamma }\qquad }$	(both occurrences of ${\displaystyle \alpha }$ are lower; ${\displaystyle A_{\alpha }{}^{\alpha \gamma }}$ would be fine)
${\displaystyle A_{\alpha \gamma }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }}$	(${\displaystyle \gamma }$ occurs twice as a lower index; ${\displaystyle A_{\alpha \gamma }{}^{\gamma }B^{\alpha }}$ or ${\displaystyle A_{\alpha \delta }{}^{\gamma }B^{\alpha }C_{\gamma }{}^{\beta }}$ would be fine).

The reason for excluding such formulae is that although these quantities could be computed as arrays of numbers, they would not in general transform as tensors under a change of basis.

If a tensor has a list of all upper or lower indices, one shorthand is to use a capital letter for the list:^[9]

${\displaystyle A_{i_{1}\cdots i_{n}}B^{i_{1}\cdots i_{n}j_{1}\cdots j_{m}}C_{j_{1}\cdots j_{m}}\equiv A_{I}B^{IJ}C_{J},}$

where I = i₁ i₂ ⋅⋅⋅ i_n and J = j₁ j₂ ⋅⋅⋅ j_m.

A pair of vertical bars | ⋅ | around a set of all-upper indices or all-lower indices (but not both), associated with contraction with another set of indices when the expression is completely antisymmetric in each of the two sets of indices:^[10]

${\displaystyle A_{|\alpha \beta \gamma |\cdots }B^{\alpha \beta \gamma \cdots }=A_{\alpha \beta \gamma \cdots }B^{|\alpha \beta \gamma |\cdots }=\sum _{\alpha <\beta <\gamma }A_{\alpha \beta \gamma \cdots }B^{\alpha \beta \gamma \cdots }}$

means a restricted sum over index values, where each index is constrained to being strictly less than the next. More than one group can be summed in this way, for example:

${\displaystyle {\begin{aligned}&A_{|\alpha \beta \gamma |}{}^{|\delta \epsilon \cdots \lambda |}B^{\alpha \beta \gamma }{}_{\delta \epsilon \cdots \lambda |\mu \nu \cdots \zeta |}C^{\mu \nu \cdots \zeta }\\[3pt]={}&\sum _{\alpha <\beta <\gamma }~\sum _{\delta <\epsilon <\cdots <\lambda }~\sum _{\mu <\nu <\cdots <\zeta }A_{\alpha \beta \gamma }{}^{\delta \epsilon \cdots \lambda }B^{\alpha \beta \gamma }{}_{\delta \epsilon \cdots \lambda \mu \nu \cdots \zeta }C^{\mu \nu \cdots \zeta }\end{aligned}}}$

When using multi-index notation, an underarrow is placed underneath the block of indices:^[11]

${\displaystyle A_{\underset {\rightharpoondown }{P}}{}^{\underset {\rightharpoondown }{Q}}B^{P}{}_{Q{\underset {\rightharpoondown }{R}}}C^{R}=\sum _{\underset {\rightharpoondown }{P}}\sum _{\underset {\rightharpoondown }{Q}}\sum _{\underset {\rightharpoondown }{R}}A_{P}{}^{Q}B^{P}{}_{QR}C^{R}}$

where

${\displaystyle {\underset {\rightharpoondown }{P}}=|\alpha \beta \gamma |\,,\quad {\underset {\rightharpoondown }{Q}}=|\delta \epsilon \cdots \lambda |\,,\quad {\underset {\rightharpoondown }{R}}=|\mu \nu \cdots \zeta |}$

By contracting an index with a non-singular metric tensor, the type of a tensor can be changed, converting a lower index to an upper index or vice versa:

${\displaystyle B^{\gamma }{}_{\beta \cdots }=g^{\gamma \alpha }A_{\alpha \beta \cdots }\quad {\text{and}}\quad A_{\alpha \beta \cdots }=g_{\alpha \gamma }B^{\gamma }{}_{\beta \cdots }}$

The base symbol in many cases is retained (e.g. using A where B appears here), and when there is no ambiguity, repositioning an index may be taken to imply this operation.

This table summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a passive transformation between bases, with the components of each basis set in terms of the other reflected in the first column. The barred indices refer to the final coordinate system after the transformation.^[12]

The Kronecker delta is used, see also below.

	Basis transformation	Component transformation	Invariance
Covector, covariant vector, 1-form	${\displaystyle \omega ^{\bar {\alpha }}=L_{\beta }{}^{\bar {\alpha }}\omega ^{\beta }}$	${\displaystyle a_{\bar {\alpha }}=a_{\gamma }L^{\gamma }{}_{\bar {\alpha }}}$	${\displaystyle a_{\bar {\alpha }}\omega ^{\bar {\alpha }}=a_{\gamma }L^{\gamma }{}_{\bar {\alpha }}L_{\beta }{}^{\bar {\alpha }}\omega ^{\beta }=a_{\gamma }\delta ^{\gamma }{}_{\beta }\omega ^{\beta }=a_{\beta }\omega ^{\beta }}$
Vector, contravariant vector	${\displaystyle e_{\bar {\alpha }}=e_{\gamma }L_{\bar {\alpha }}{}^{\gamma }}$	${\displaystyle u^{\bar {\alpha }}=L^{\bar {\alpha }}{}_{\beta }u^{\beta }}$	${\displaystyle e_{\bar {\alpha }}u^{\bar {\alpha }}=e_{\gamma }L_{\bar {\alpha }}{}^{\gamma }L^{\bar {\alpha }}{}_{\beta }u^{\beta }=e_{\gamma }\delta ^{\gamma }{}_{\beta }u^{\beta }=e_{\gamma }u^{\gamma }}$

Tensors are equal if and only if every corresponding component is equal; e.g., tensor A equals tensor B if and only if

${\displaystyle A^{\alpha }{}_{\beta \gamma }=B^{\alpha }{}_{\beta \gamma }}$

for all α, β, γ. Consequently, there are facets of the notation that are useful in checking that an equation makes sense (an analogous procedure to dimensional analysis).

Indices not involved in contractions are called free indices. Indices used in contractions are termed dummy indices, or summation indices.

The components of tensors (like A^α, B_β^γ etc.) are just real numbers. Since the indices take various integer values to select specific components of the tensors, a single tensor equation represents many ordinary equations. If a tensor equality has n free indices, and if the dimensionality of the underlying vector space is m, the equality represents mⁿ equations: each index takes on every value of a specific set of values.

For instance, if

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }=T^{\alpha }{}_{\beta }{}_{\delta }}$

is in four dimensions (that is, each index runs from 0 to 3 or from 1 to 4), then because there are three free indices (α, β, δ), there are 4³ = 64 equations. Three of these are:

${\displaystyle {\begin{aligned}A^{0}B_{1}{}^{0}C_{00}+A^{0}B_{1}{}^{1}C_{10}+A^{0}B_{1}{}^{2}C_{20}+A^{0}B_{1}{}^{3}C_{30}+D^{0}{}_{1}{}E_{0}&=T^{0}{}_{1}{}_{0}\\A^{1}B_{0}{}^{0}C_{00}+A^{1}B_{0}{}^{1}C_{10}+A^{1}B_{0}{}^{2}C_{20}+A^{1}B_{0}{}^{3}C_{30}+D^{1}{}_{0}{}E_{0}&=T^{1}{}_{0}{}_{0}\\A^{1}B_{2}{}^{0}C_{02}+A^{1}B_{2}{}^{1}C_{12}+A^{1}B_{2}{}^{2}C_{22}+A^{1}B_{2}{}^{3}C_{32}+D^{1}{}_{2}{}E_{2}&=T^{1}{}_{2}{}_{2}.\end{aligned}}}$

This illustrates the compactness and efficiency of using index notation: many equations which all share a similar structure can be collected into one simple tensor equation.

Replacing any index symbol throughout by another leaves the tensor equation unchanged (provided there is no conflict with other symbols already used). This can be useful when manipulating indices, such as using index notation to verify vector calculus identities or identities of the Kronecker delta and Levi-Civita symbol (see also below). An example of a correct change is:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }\rightarrow A^{\lambda }B_{\beta }{}^{\mu }C_{\mu \delta }+D^{\lambda }{}_{\beta }{}E_{\delta }\,,}$

whereas an erroneous change is:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }\nrightarrow A^{\lambda }B_{\beta }{}^{\gamma }C_{\mu \delta }+D^{\alpha }{}_{\beta }{}E_{\delta }\,.}$

In the first replacement, λ replaced α and μ replaced γ everywhere, so the expression still has the same meaning. In the second, λ did not fully replace α, and μ did not fully replace γ (incidentally, the contraction on the γ index became a tensor product), which is entirely inconsistent for reasons shown next.

The free indices in a tensor expression always appear in the same (upper or lower) position throughout every term, and in a tensor equation the free indices are the same on each side. Dummy indices (which implies a summation over that index) need not be the same, for example:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D^{\alpha }{}_{\delta }E_{\beta }=T^{\alpha }{}_{\beta }{}_{\delta }}$

as for an erroneous expression:

${\displaystyle A^{\alpha }B_{\beta }{}^{\gamma }C_{\gamma \delta }+D_{\alpha }{}_{\beta }{}^{\gamma }E^{\delta }.}$

In other words, non-repeated indices must be of the same type in every term of the equation. In the above identity, α, β, δ line up throughout and γ occurs twice in one term due to a contraction (once as an upper index and once as a lower index), and thus it is a valid expression. In the invalid expression, while β lines up, α and δ do not, and γ appears twice in one term (contraction) and once in another term, which is inconsistent.

When applying a rule to a number of indices (differentiation, symmetrization etc., shown next), the bracket or punctuation symbols denoting the rules are only shown on one group of the indices to which they apply.

If the brackets enclose covariant indices – the rule applies only to all covariant indices enclosed in the brackets, not to any contravariant indices which happen to be placed intermediately between the brackets.

Similarly if brackets enclose contravariant indices – the rule applies only to all enclosed contravariant indices, not to intermediately placed covariant indices.

Parentheses, ( ), around multiple indices denotes the symmetrized part of the tensor. When symmetrizing p indices using σ to range over permutations of the numbers 1 to p, one takes a sum over the permutations of those indices α_σ(i) for i = 1, 2, 3, ..., p, and then divides by the number of permutations:

${\displaystyle A_{(\alpha _{1}\alpha _{2}\cdots \alpha _{p})\alpha _{p+1}\cdots \alpha _{q}}={\dfrac {1}{p!}}\sum _{\sigma }A_{\alpha _{\sigma (1)}\cdots \alpha _{\sigma (p)}\alpha _{p+1}\cdots \alpha _{q}}\,.}$

For example, two symmetrizing indices mean there are two indices to permute and sum over:

${\displaystyle A_{(\alpha \beta )\gamma \cdots }={\dfrac {1}{2!}}\left(A_{\alpha \beta \gamma \cdots }+A_{\beta \alpha \gamma \cdots }\right)}$

while for three symmetrizing indices, there are three indices to sum over and permute:

${\displaystyle A_{(\alpha \beta \gamma )\delta \cdots }={\dfrac {1}{3!}}\left(A_{\alpha \beta \gamma \delta \cdots }+A_{\gamma \alpha \beta \delta \cdots }+A_{\beta \gamma \alpha \delta \cdots }+A_{\alpha \gamma \beta \delta \cdots }+A_{\gamma \beta \alpha \delta \cdots }+A_{\beta \alpha \gamma \delta \cdots }\right)}$

The symmetrization is distributive over addition;

${\displaystyle A_{(\alpha }\left(B_{\beta )\gamma \cdots }+C_{\beta )\gamma \cdots }\right)=A_{(\alpha }B_{\beta )\gamma \cdots }+A_{(\alpha }C_{\beta )\gamma \cdots }}$

Indices are not part of the symmetrization when they are:

• not on the same level, for example;

  ${\displaystyle A_{(\alpha }B^{\beta }{}_{\gamma )}={\dfrac {1}{2!}}\left(A_{\alpha }B^{\beta }{}_{\gamma }+A_{\gamma }B^{\beta }{}_{\alpha }\right)}$

• within the parentheses and between vertical bars (i.e. |⋅⋅⋅|), modifying the previous example;

  ${\displaystyle A_{(\alpha }B_{|\beta |}{}_{\gamma )}={\dfrac {1}{2!}}\left(A_{\alpha }B_{\beta \gamma }+A_{\gamma }B_{\beta \alpha }\right)}$

Here the α and γ indices are symmetrized, β is not.

Square brackets, [ ], around multiple indices denotes the antisymmetrized part of the tensor. For p antisymmetrizing indices – the sum over the permutations of those indices α_σ(i) multiplied by the signature of the permutation sgn(σ) is taken, then divided by the number of permutations:

${\displaystyle {\begin{aligned}&A_{[\alpha _{1}\cdots \alpha _{p}]\alpha _{p+1}\cdots \alpha _{q}}\\[3pt]={}&{\dfrac {1}{p!}}\sum _{\sigma }\operatorname {sgn}(\sigma )A_{\alpha _{\sigma (1)}\cdots \alpha _{\sigma (p)}\alpha _{p+1}\cdots \alpha _{q}}\\={}&\delta _{\alpha _{1}\cdots \alpha _{p}}^{\beta _{1}\dots \beta _{p}}A_{\beta _{1}\cdots \beta _{p}\alpha _{p+1}\cdots \alpha _{q}}\\\end{aligned}}}$

where δβ1⋅⋅⋅βp
α1⋅⋅⋅αp is the generalized Kronecker delta of degree 2p, with scaling as defined below.

For example, two antisymmetrizing indices imply:

${\displaystyle A_{[\alpha \beta ]\gamma \cdots }={\dfrac {1}{2!}}\left(A_{\alpha \beta \gamma \cdots }-A_{\beta \alpha \gamma \cdots }\right)}$

while three antisymmetrizing indices imply:

${\displaystyle A_{[\alpha \beta \gamma ]\delta \cdots }={\dfrac {1}{3!}}\left(A_{\alpha \beta \gamma \delta \cdots }+A_{\gamma \alpha \beta \delta \cdots }+A_{\beta \gamma \alpha \delta \cdots }-A_{\alpha \gamma \beta \delta \cdots }-A_{\gamma \beta \alpha \delta \cdots }-A_{\beta \alpha \gamma \delta \cdots }\right)}$

as for a more specific example, if F represents the electromagnetic tensor, then the equation

${\displaystyle 0=F_{[\alpha \beta ,\gamma ]}={\dfrac {1}{3!}}\left(F_{\alpha \beta ,\gamma }+F_{\gamma \alpha ,\beta }+F_{\beta \gamma ,\alpha }-F_{\beta \alpha ,\gamma }-F_{\alpha \gamma ,\beta }-F_{\gamma \beta ,\alpha }\right)\,}$

represents Gauss's law for magnetism and Faraday's law of induction.

As before, the antisymmetrization is distributive over addition;

${\displaystyle A_{[\alpha }\left(B_{\beta ]\gamma \cdots }+C_{\beta ]\gamma \cdots }\right)=A_{[\alpha }B_{\beta ]\gamma \cdots }+A_{[\alpha }C_{\beta ]\gamma \cdots }}$

As with symmetrization, indices are not antisymmetrized when they are:

• not on the same level, for example;

  ${\displaystyle A_{[\alpha }B^{\beta }{}_{\gamma ]}={\dfrac {1}{2!}}\left(A_{\alpha }B^{\beta }{}_{\gamma }-A_{\gamma }B^{\beta }{}_{\alpha }\right)}$

• within the square brackets and between vertical bars (i.e. |⋅⋅⋅|), modifying the previous example;

  ${\displaystyle A_{[\alpha }B_{|\beta |}{}_{\gamma ]}={\dfrac {1}{2!}}\left(A_{\alpha }B_{\beta \gamma }-A_{\gamma }B_{\beta \alpha }\right)}$

Here the α and γ indices are antisymmetrized, β is not.

Any tensor can be written as the sum of its symmetric and antisymmetric parts on two indices:

${\displaystyle A_{\alpha \beta \gamma \cdots }=A_{(\alpha \beta )\gamma \cdots }+A_{[\alpha \beta ]\gamma \cdots }}$

as can be seen by adding the above expressions for A_{(αβ)γ⋅⋅⋅} and A_{[αβ]γ⋅⋅⋅}. This does not hold for other than two indices.

For compactness, derivatives may be indicated by adding indices after a comma or semicolon.^[13][14]

While most of the expressions of the Ricci calculus are valid for arbitrary bases, the expressions involving partial derivatives of tensor components with respect to coordinates apply only with a coordinate basis: a basis that is defined through differentiation with respect to the coordinates. Coordinates are typically denoted by x^μ, but do not in general form the components of a vector. In flat spacetime with linear coordinatization, a tuple of differences in coordinates, Δx^μ, can be treated as a contravariant vector. With the same constraints on the space and on the choice of coordinate system, the partial derivatives with respect to the coordinates yield a result that is effectively covariant. Aside from use in this special case, the partial derivatives of components of tensors do not in general transform covariantly, but are useful in building expressions that are covariant, albeit still with a coordinate basis if the partial derivatives are explicitly used, as with the covariant, exterior and Lie derivatives below.

To indicate partial differentiation of the components of a tensor field with respect to a coordinate variable x^γ, a comma is placed before an appended lower index of the coordinate variable.

${\displaystyle A_{\alpha \beta \cdots ,\gamma }={\dfrac {\partial }{\partial x^{\gamma }}}A_{\alpha \beta \cdots }}$

This may be repeated (without adding further commas):

${\displaystyle A_{\alpha _{1}\alpha _{2}\cdots \alpha _{p}\,,\,\alpha _{p+1}\cdots \alpha _{q}}={\dfrac {\partial }{\partial x^{\alpha _{q}}}}\cdots {\dfrac {\partial }{\partial x^{\alpha _{p+2}}}}{\dfrac {\partial }{\partial x^{\alpha _{p+1}}}}A_{\alpha _{1}\alpha _{2}\cdots \alpha _{p}}.}$

These components do not transform covariantly, unless the expression being differentiated is a scalar. This derivative is characterized by the product rule and the derivatives of the coordinates

${\displaystyle x^{\alpha }{}_{,\gamma }=\delta _{\gamma }^{\alpha },}$

where δ is the Kronecker delta.

The covariant derivative is only defined if a connection is defined. For any tensor field, a semicolon ( ; ) placed before an appended lower (covariant) index indicates covariant differentiation. Less common alternatives to the semicolon include a forward slash ( / )^[15] or in three-dimensional curved space a single vertical bar ( | ).^[16]

The covariant derivative of a scalar function, a contravariant vector and a covariant vector are:

${\displaystyle f_{;\beta }=f_{,\beta }}$

${\displaystyle A^{\alpha }{}_{;\beta }=A^{\alpha }{}_{,\beta }+\Gamma ^{\alpha }{}_{\gamma \beta }A^{\gamma }}$

${\displaystyle A_{\alpha ;\beta }=A_{\alpha ,\beta }-\Gamma ^{\gamma }{}_{\alpha \beta }A_{\gamma }\,,}$

where Γ^α_γβ are the connection coefficients.

For an arbitrary tensor:^[17]

${\displaystyle {\begin{aligned}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s};\gamma }&\\=T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s},\gamma }&+\,\Gamma ^{\alpha _{1}}{}_{\delta \gamma }T^{\delta \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}+\cdots +\Gamma ^{\alpha _{r}}{}_{\delta \gamma }T^{\alpha _{1}\cdots \alpha _{r-1}\delta }{}_{\beta _{1}\cdots \beta _{s}}\\&-\,\Gamma ^{\delta }{}_{\beta _{1}\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\delta \beta _{2}\cdots \beta _{s}}-\cdots -\Gamma ^{\delta }{}_{\beta _{s}\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\delta }\,.\end{aligned}}}$

An alternative notation for the covariant derivative of any tensor is the subscripted nabla symbol ∇_β. For the case of a vector field A^α:^[18]

${\displaystyle \nabla _{\beta }A^{\alpha }=A^{\alpha }{}_{;\beta }\,.}$

The covariant formulation of the directional derivative of any tensor field along a vector v^γ may be expressed as its contraction with the covariant derivative, e.g.:

${\displaystyle v^{\gamma }A_{\alpha ;\gamma }\,.}$

The components of this derivative of a tensor field transform covariantly, and hence form another tensor field, despite subexpressions (the partial derivative and the connection coefficients) separately not transforming covariantly.

This derivative is characterized by the product rule:

${\displaystyle (A^{\alpha }{}_{\beta \cdots }B^{\gamma }{}_{\delta \cdots })_{;\epsilon }=A^{\alpha }{}_{\beta \cdots ;\epsilon }B^{\gamma }{}_{\delta \cdots }+A^{\alpha }{}_{\beta \cdots }B^{\gamma }{}_{\delta \cdots ;\epsilon }\,.}$

A Koszul connection on the tangent bundle of a differentiable manifold is called an affine connection.

A connection is a metric connection when the covariant derivative of the metric tensor vanishes:

${\displaystyle g_{\mu \nu ;\xi }=0\,.}$

An affine connection that is also a metric connection is called a Riemannian connection. A Riemannian connection that is torsion-free (i.e., for which the torsion tensor vanishes: T^α_βγ = 0) is a Levi-Civita connection.

The Γ^α_βγ for a Levi-Civita connection in a coordinate basis are called Christoffel symbols of the second kind.

The exterior derivative of a totally antisymmetric type (0, s) tensor field with components A_{α1⋅⋅⋅αs} (also called a differential form) is a derivative that is covariant under basis transformations. It does not depend on either a metric tensor or a connection: it requires only the structure of a differentiable manifold. In a coordinate basis, it may be expressed as the antisymmetrization of the partial derivatives of the tensor components:^{[3]: 232–233}

${\displaystyle (\mathrm {d} A)_{\gamma \alpha _{1}\cdots \alpha _{s}}={\frac {\partial }{\partial x^{[\gamma }}}A_{\alpha _{1}\cdots \alpha _{s}]}=A_{[\alpha _{1}\cdots \alpha _{s},\gamma ]}.}$

This derivative is not defined on any tensor field with contravariant indices or that is not totally antisymmetric. It is characterized by a graded product rule.

The Lie derivative is another derivative that is covariant under basis transformations. Like the exterior derivative, it does not depend on either a metric tensor or a connection. The Lie derivative of a type (r, s) tensor field T along (the flow of) a contravariant vector field X^ρ may be expressed using a coordinate basis as^[19]

${\displaystyle {\begin{aligned}({\mathcal {L}}_{X}T)^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}&\\=X^{\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s},\gamma }&-\,X^{\alpha _{1}}{}_{,\gamma }T^{\gamma \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}-\cdots -X^{\alpha _{r}}{}_{,\gamma }T^{\alpha _{1}\cdots \alpha _{r-1}\gamma }{}_{\beta _{1}\cdots \beta _{s}}\\&+\,X^{\gamma }{}_{,\beta _{1}}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\gamma \beta _{2}\cdots \beta _{s}}+\cdots +X^{\gamma }{}_{,\beta _{s}}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\gamma }\,.\end{aligned}}}$

This derivative is characterized by the product rule and the fact that the Lie derivative of a contravariant vector field along itself is zero:

${\displaystyle ({\mathcal {L}}_{X}X)^{\alpha }=X^{\gamma }X^{\alpha }{}_{,\gamma }-X^{\alpha }{}_{,\gamma }X^{\gamma }=0\,.}$

The Kronecker delta is like the identity matrix when multiplied and contracted:

${\displaystyle {\begin{aligned}\delta _{\beta }^{\alpha }\,A^{\beta }&=A^{\alpha }\\\delta _{\nu }^{\mu }\,B_{\mu }&=B_{\nu }.\end{aligned}}}$

The components δ^α
_β are the same in any basis and form an invariant tensor of type (1, 1), i.e. the identity of the tangent bundle over the identity mapping of the base manifold, and so its trace is an invariant.^[20] Its trace is the dimensionality of the space; for example, in four-dimensional spacetime,

${\displaystyle \delta _{\rho }^{\rho }=\delta _{0}^{0}+\delta _{1}^{1}+\delta _{2}^{2}+\delta _{3}^{3}=4.}$

The Kronecker delta is one of the family of generalized Kronecker deltas. The generalized Kronecker delta of degree 2p may be defined in terms of the Kronecker delta by (a common definition includes an additional multiplier of p! on the right):

${\displaystyle \delta _{\beta _{1}\cdots \beta _{p}}^{\alpha _{1}\cdots \alpha _{p}}=\delta _{\beta _{1}}^{[\alpha _{1}}\cdots \delta _{\beta _{p}}^{\alpha _{p}]},}$

and acts as an antisymmetrizer on p indices:

${\displaystyle \delta _{\beta _{1}\cdots \beta _{p}}^{\alpha _{1}\cdots \alpha _{p}}\,A^{\beta _{1}\cdots \beta _{p}}=A^{[\alpha _{1}\cdots \alpha _{p}]}.}$

An affine connection has a torsion tensor T^α_βγ:

${\displaystyle T^{\alpha }{}_{\beta \gamma }=\Gamma ^{\alpha }{}_{\beta \gamma }-\Gamma ^{\alpha }{}_{\gamma \beta }-\gamma ^{\alpha }{}_{\beta \gamma },}$

where γ^α_βγ are given by the components of the Lie bracket of the local basis, which vanish when it is a coordinate basis.

For a Levi-Civita connection this tensor is defined to be zero, which for a coordinate basis gives the equations

${\displaystyle \Gamma ^{\alpha }{}_{\beta \gamma }=\Gamma ^{\alpha }{}_{\gamma \beta }.}$

If this tensor is defined as

${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\Gamma ^{\rho }{}_{\nu \sigma ,\mu }-\Gamma ^{\rho }{}_{\mu \sigma ,\nu }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }\,,}$

then it is the commutator of the covariant derivative with itself:^[21][22]

${\displaystyle A_{\nu ;\rho \sigma }-A_{\nu ;\sigma \rho }=A_{\beta }R^{\beta }{}_{\nu \rho \sigma }\,,}$

since the connection is torsionless, which means that the torsion tensor vanishes.

This can be generalized to get the commutator for two covariant derivatives of an arbitrary tensor as follows:

${\displaystyle {\begin{aligned}T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s};\gamma \delta }&-T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s};\delta \gamma }\\&\!\!\!\!\!\!\!\!\!\!=-R^{\alpha _{1}}{}_{\rho \gamma \delta }T^{\rho \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}-\cdots -R^{\alpha _{r}}{}_{\rho \gamma \delta }T^{\alpha _{1}\cdots \alpha _{r-1}\rho }{}_{\beta _{1}\cdots \beta _{s}}\\&+R^{\sigma }{}_{\beta _{1}\gamma \delta }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\sigma \beta _{2}\cdots \beta _{s}}+\cdots +R^{\sigma }{}_{\beta _{s}\gamma \delta }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\sigma }\,\end{aligned}}}$

which are often referred to as the Ricci identities.^[23]

The metric tensor g_αβ is used for lowering indices and gives the length of any space-like curve

${\displaystyle {\text{length}}=\int _{y_{1}}^{y_{2}}{\sqrt {g_{\alpha \beta }{\frac {dx^{\alpha }}{d\gamma }}{\frac {dx^{\beta }}{d\gamma }}}}\,d\gamma \,,}$

where γ is any smooth strictly monotone parameterization of the path. It also gives the duration of any time-like curve

${\displaystyle {\text{duration}}=\int _{t_{1}}^{t_{2}}{\sqrt {{\frac {-1}{c^{2}}}g_{\alpha \beta }{\frac {dx^{\alpha }}{d\gamma }}{\frac {dx^{\beta }}{d\gamma }}}}\,d\gamma \,,}$

where γ is any smooth strictly monotone parameterization of the trajectory. See also Line element.

The inverse matrix g^αβ of the metric tensor is another important tensor, used for raising indices:

${\displaystyle g^{\alpha \beta }g_{\beta \gamma }=\delta _{\gamma }^{\alpha }\,.}$

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