A derivative in Differential Geometry
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.
Functions, tensor fields and forms can be differentiated with respect to a vector field. If "T" is a tensor field and "X" is a vector field, then the Lie derivative of "T" with respect to "X" is denoted formula_1. The differential operator formula_2 is a derivation of the algebra of tensor fields of the underlying manifold.
The Lie derivative commutes with contraction and the exterior derivative on differential forms.
Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function.
The Lie derivative of a vector field "Y" with respect to another vector field "X" is known as the "Lie bracket" of "X" and "Y", and is often denoted ["X","Y"] instead of formula_3. The space of vector fields forms a Lie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra, due to the identity
formula_4
valid for any vector fields "X" and "Y" and any tensor field "T".
Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on "M", the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation in Lie group theory.
Generalisations exist for spinor fields, fibre bundles with a connection and vector-valued differential forms.
Motivation.
A 'naïve' attempt to define the derivative of a tensor field with respect to a vector field would be to take the components of the tensor field and take the directional derivative of each component with respect to the vector field. However, this definition is undesirable because it is not invariant under changes of coordinate system, e.g. the naive derivative expressed in polar or spherical coordinates differs from the naive derivative of the components in Cartesian coordinates. On an abstract manifold such a definition is meaningless and ill defined.
In differential geometry, there are three main coordinate independent notions of differentiation of tensor fields:
The main difference between the Lie derivative and a derivative with respect to a connection is that the latter derivative of a tensor field with respect to a tangent vector is well-defined even if it is not specified how to extend that tangent vector to a vector field. However, a connection requires the choice of an additional geometric structure (e.g. a Riemannian metric in the case of Levi-Civita connection, or just an abstract connection) on the manifold. In contrast, when taking a Lie derivative, no additional structure on the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector field "X" at a point "p" depends on the value of "X" in a neighborhood of "p", not just at "p" itself. Finally, the exterior derivative of differential forms does not require any additional choices, but is only a well defined derivative of differential forms (including functions), thus excluding vectors and other tensors that are not purely differential forms.
The idea of Lie derivatives is to use a vector field to define a notion of transport (Lie transport). A smooth vector field defines a smooth flow on the manifold, which allows vectors to be transported between two points on the same line of flow (This contrasts with connections, which allows transport between arbitrary points). Intuitively, a vector formula_5 based at point formula_6 is transported by flowing its base point to formula_7, while flowing its tip point formula_8 to formula_9.
Definition.
The Lie derivative may be defined in several equivalent ways. To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors.
The (Lie) derivative of a function.
Defining the derivative of a function formula_10 on a manifold is problematic because the difference quotient formula_11 cannot be determined while the displacement formula_12 is undefined.
The Lie derivative of a function formula_13 with respect to a vector field formula_14 at a point formula_15 is the function
formula_16
where formula_17 is the point to which the flow defined by the vector field formula_14 maps the point formula_6 at time instant formula_20 In the vicinity of formula_21 formula_17 is the unique solution of the system
formula_23
of first-order autonomous (i.e. time-independent) differential equations, with formula_24
Setting formula_25 identifies the Lie derivative of a function with the directional derivative, which is also denoted by formula_26. 
The Lie derivative of a vector field.
If "X" and "Y" are both vector fields, then the Lie derivative of "Y" with respect to "X" is also known as the Lie bracket of "X" and "Y", and is sometimes denoted formula_27. There are several approaches to defining the Lie bracket, all of which are equivalent. We list two definitions here, corresponding to the two definitions of a vector field given above:
The Lie derivative of a tensor field.
Definition in terms of flows.
The Lie derivative is the speed with which the tensor field changes under the space deformation caused by the flow.
Formally, given a differentiable (time-independent) vector field formula_14 on a smooth manifold formula_29 let formula_30 be the corresponding local flow. Since formula_31 is a local diffeomorphism for each formula_32, it gives rise to a pullback of tensor fields. For covariant tensors, this is just the multi-linear extension of the pullback map
formula_33
For contravariant tensors, one extends the inverse
formula_34
of the differential formula_35. For every formula_36 there is, consequently, a tensor field formula_37 of the same type as formula_38's.
If formula_38 is an formula_40- or formula_41-type tensor field, then the Lie derivative formula_42 of formula_38 along a vector field formula_14 is defined at point formula_15 to be
formula_46
The resulting tensor field formula_47 is of the same type as formula_38's.
More generally, for every smooth 1-parameter family formula_49of diffeomorphisms that integrate a vector field formula_50 in the sense that formula_51, one hasformula_52
Algebraic definition.
We now give an algebraic definition. The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms:
Axiom 1. The Lie derivative of a function is equal to the directional derivative of the function. This fact is often expressed by the formula
formula_53
Axiom 2. The Lie derivative obeys the following version of Leibniz's rule: For any tensor fields "S" and "T", we have
formula_54
Axiom 3. The Lie derivative obeys the Leibniz rule with respect to contraction:
formula_55
Axiom 4. The Lie derivative commutes with exterior derivative on functions:
formula_56
If these axioms hold, then applying the Lie derivative formula_57 to the relation formula_58 shows that
formula_59
which is one of the standard definitions for the Lie bracket.
The Lie derivative acting on a differential form is the anticommutator of the interior product with the exterior derivative. So if α is a differential form,
formula_60
This follows easily by checking that the expression commutes with exterior derivative, is a derivation (being an anticommutator of graded derivations) and does the right thing on functions. This is Cartan's magic formula. See interior product for details.
Explicitly, let "T" be a tensor field of type ("p", "q"). Consider "T" to be a differentiable multilinear map of smooth sections "α"1, "α"2, ..., "α""p" of the cotangent bundle "T"∗"M" and of sections "X"1, "X"2, ..., "X"q of the tangent bundle "TM", written "T"("α"1, "α"2, ..., "X"1, "X"2, ...) into R. Define the Lie derivative of "T" along "Y" by the formula
formula_61
formula_62
formula_63
The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the Leibniz rule for differentiation. The Lie derivative commutes with the contraction.
The Lie derivative of a differential form.
A particularly important class of tensor fields is the class of differential forms. The restriction of the Lie derivative to the space of differential forms is closely related to the exterior derivative. Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of an interior product, after which the relationships falls out as an identity known as Cartan's formula. Cartan's formula can also be used as a definition of the Lie derivative on the space of differential forms.
Let "M" be a manifold and "X" a vector field on "M". Let formula_64 be a "k"-form, i.e., for each formula_15, formula_66 is an alternating multilinear map from formula_67 to the real numbers. The interior product of "X" and "ω" is the ("k" − 1)-form formula_68 defined as
formula_69
The differential form formula_68 is also called the contraction of "ω" with "X", and
formula_71
is a formula_72-antiderivation where formula_72 is the wedge product on differential forms. That is, formula_74 is R-linear, and
formula_75
for formula_64 and η another differential form. Also, for a function formula_77, that is, a real- or complex-valued function on "M", one has
formula_78
where formula_79 denotes the product of "f" and "X".
The relationship between exterior derivatives and Lie derivatives can then be summarized as follows. First, since the Lie derivative of a function "f" with respect to a vector field "X" is the same as the directional derivative "X"("f"), it is also the same as the contraction of the exterior derivative of "f" with "X":
formula_80
For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in "X":
formula_81
This identity is known variously as Cartan formula, Cartan homotopy formula or Cartan's magic formula. See interior product for details. The Cartan formula can be used as a definition of the Lie derivative of a differential form. Cartan's formula shows in particular that
formula_82
The Lie derivative also satisfies the relation
formula_83
Coordinate expressions.
In local coordinate notation, for a type ("r", "s") tensor field formula_38, the Lie derivative along formula_14 is
formula_86
here, the notation formula_87 means taking the partial derivative with respect to the coordinate formula_88. Alternatively, if we are using a torsion-free connection (e.g., the Levi Civita connection), then the partial derivative formula_89 can be replaced with the covariant derivative which means replacing formula_90 with (by abuse of notation) formula_91 where the formula_92 are the Christoffel coefficients.
The Lie derivative of a tensor is another tensor of the same type, i.e., even though the individual terms in the expression depend on the choice of coordinate system, the expression as a whole results in a tensor
formula_93
which is independent of any coordinate system and of the same type as formula_38.
The definition can be extended further to tensor densities. If "T" is a tensor density of some real number valued weight "w" (e.g. the volume density of weight 1), then its Lie derivative is a tensor density of the same type and weight.
formula_95
Notice the new term at the end of the expression.
For a linear connection formula_96, the Lie derivative along formula_14 is
formula_98
Examples.
For clarity we now show the following examples in local coordinate notation.
For a scalar field formula_99 we have:
formula_100.
Hence for the scalar field formula_101 and the vector field formula_102 the corresponding Lie derivative becomes 
formula_103
For an example of higher rank differential form, consider the 2-form formula_104 and the vector field formula_14 from the previous example. Then,
formula_106
Some more abstract examples.
formula_107.
Hence for a covector field, i.e., a differential form, formula_108 we have:
formula_109
The coefficient of the last expression is the local coordinate expression of the Lie derivative. 
For a covariant rank 2 tensor field formula_110 we have:
formula_111
If formula_112 is the symmetric metric tensor, it is parallel with respect to the Levi-Civita connection (aka covariant derivative), and it becomes fruitful to use the connection. This has the effect of replacing all derivatives with covariant derivatives, giving
formula_113
Properties.
The Lie derivative has a number of properties. Let formula_114 be the algebra of functions defined on the manifold "M". Then
formula_115
is a derivation on the algebra formula_114. That is,
formula_57 is R-linear and
formula_118
Similarly, it is a derivation on formula_119 where formula_120 is the set of vector fields on "M":
formula_121
which may also be written in the equivalent notation
formula_122
where the tensor product symbol formula_123 is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold.
Additional properties are consistent with that