Lie derivative

In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field, 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. The differential operator 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 instead of. 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

Proof of the identity

:
valid for any vector fields X and Y and any tensor field T.
Considering vector fields as infinitesimal generators of flows on M, the Lie derivative is the Lie algebra representation#Infinitesimal [Lie group representations|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 differential forms">pushforward (differential)">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:

Lie derivatives,
derivatives with respect to connections,
the exterior derivative of totally antisymmetric covariant tensors, i.e. differential forms.

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 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, 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. 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. Intuitively, a vector based at point is transported by flowing its base point to, while flowing its tip point to.

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 on a manifold takes care because the difference quotient cannot be determined while the displacement is undefined.
The Lie derivative of a function with respect to a vector field at a point is the function
where is the point to which the flow defined by the vector field maps the point at time instant In the vicinity of is the unique solution of the system
of first-order autonomous differential equations, with
Setting identifies the Lie derivative of a function with the directional derivative, which is also denoted by.

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. 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 vector field on a smooth manifold let be the corresponding local flow. Since is a local diffeomorphism for each, it gives rise to a pullback of tensor fields. For covariant tensors, this is just the multi-linear extension of the pullback map
For contravariant tensors, one extends the inverse
of the differential. For every there is, consequently, a tensor field of the same type as 's.
If is an - or -type tensor field, then the Lie derivative of along a vector field is defined at point to be
The resulting tensor field is of the same type as 's.
More generally, for every smooth 1-parameter family of diffeomorphisms that integrate a vector field in the sense that, one has

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:
Using the first and third axioms, applying the Lie derivative to shows that
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,
This follows easily by checking that the expression commutes with exterior derivative, is a derivation 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. Consider T to be a differentiable multilinear map of smooth sections α¹, α²,..., α^p of the cotangent bundle T^∗M and of sections X₁, X₂,..., X_q of the tangent bundle TM, written T into R. Define the Lie derivative of T along Y by the formula
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 be a manifold and a vector field on. Let be a -form, i.e., for each, is an alternating multilinear map from to the real numbers. The interior product of and is the -form defined as
The differential form is also called the contraction of with, and
is a -antiderivation where Exterior algebra | is the wedge product on differential forms. That is, is -linear, and
for and another differential form. Also, for a function, that is, a real- or complex-valued function on, one has
where denotes the product of and.
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, it is also the same as the contraction of the exterior derivative of f with X:
For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in X:
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
The Lie derivative also satisfies the relation

Coordinate expressions

In local coordinate notation, for a type tensor field, the Lie derivative along is
here, the notation means taking the partial derivative with respect to the coordinate. Alternatively, if we are using a torsion-free connection, then the partial derivative can be replaced with the covariant derivative which means replacing with where the 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
which is independent of any coordinate system and of the same type as.
The definition can be extended further to tensor densities. If T is a tensor density of some real number valued weight w, then its Lie derivative is a tensor density of the same type and weight.
Notice the new term at the end of the expression.
For a linear connection, the Lie derivative along is

Examples

For clarity we now show the following examples in local coordinate notation.
For a scalar field we have:
Hence for the scalar field and the vector field the corresponding Lie derivative becomes
For an example of higher rank differential form, consider the 2-form and the vector field from the previous example. Then,
Some more abstract examples.
Hence for a covector field, i.e., a differential form, we have:
The coefficient of the last expression is the local coordinate expression of the Lie derivative.
For a covariant rank 2 tensor field we have:
If is the symmetric metric tensor, it is parallel with respect to the Levi-Civita connection, and it becomes fruitful to use the connection. This has the effect of replacing all derivatives with covariant derivatives, giving

Properties

The Lie derivative has a number of properties. Let be the algebra of functions defined on the manifold. Then
is a derivation on the algebra. That is,
is -linear and
Similarly, it is a derivation on where is the set of vector fields on :
which may also be written in the equivalent notation
where the tensor product symbol 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 of the Lie bracket. Thus, for example, considered as a derivation on a vector field,
one finds the above to be just the Jacobi identity. Thus, one has the important result that the space of vector fields over M, equipped with the Lie bracket, forms a Lie algebra.
The Lie derivative also has important properties when acting on differential forms. Let α and β be two differential forms on M, and let X and Y be two vector fields. Then

where i denotes interior product defined above and it is clear whether denotes the commutator or the Lie bracket of vector fields.
Generalizations

Various generalizations of the Lie derivative play an important role in differential geometry.

The Lie derivative of a spinor field

A definition for Lie derivatives of spinors along generic spacetime vector fields, not necessarily Killing ones, on a general Riemannian manifold was already proposed in 1971 by Yvette Kosmann. Later, it was provided a geometric framework which justifies her ad hoc prescription within the general framework of Lie derivatives on fiber bundles in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for field theories.
In a given spin manifold, that is in a Riemannian manifold admitting a spin structure, the Lie derivative of a spinor field can be defined by first defining it with respect to infinitesimal isometries via the André Lichnerowicz's local expression given in 1963:
where, as is assumed to be a Killing vector field, and are Dirac matrices.
It is then possible to extend Lichnerowicz's definition to all vector fields by retaining Lichnerowicz's local expression for a generic vector field, but explicitly taking the antisymmetric part of only. More explicitly, Kosmann's local expression given in 1972 is:
where is the commutator, is exterior derivative, is the dual 1 form corresponding to under the metric and is Clifford multiplication.
It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of the connection. This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection, the dualisation of vector fields and the Clifford multiplication on the spinor bundle. Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection dependent terms cancel.
To gain a better understanding of the long-debated concept of Lie derivative of spinor fields one may refer to the original article, where the definition of a Lie derivative of spinor fields is placed in the more general framework of the theory of Lie derivatives of sections of fiber bundles and the direct approach by Y. Kosmann to the spinor case is generalized to gauge natural bundles in the form of a new geometric concept called the Kosmann lift.
As for the tensor counterpart, also for spinors the vanishing of the Lie derivative along a Killing vector implements on the spinor the symmetries encoded by that Killing vector. However, differently from tensors, from spinors it is possible to build bi-linear quantities which are tensors. A natural question that now arises is whether the vanishing of the Lie derivative along a Killing vector of a spinor is equivalent to the vanishing of the Lie derivative along the same Killing vector of all the spinor bi-linear quantities. While a spinor that is Lie-invariant implies that all its bi-linear quantities are also Lie invariant, the converse is in general not true.

Covariant Lie derivative

If we have a principal bundle over the manifold M with G as the structure group, and we pick X to be a covariant vector field as section of the tangent space of the principal bundle, then the covariant Lie derivative is just the Lie derivative with respect to X over the principal bundle.
Now, if we're given a vector field Y over M but we also have a connection over the principal bundle, we can define a vector field X over the principal bundle such that its horizontal component matches Y and its vertical component agrees with the connection. This is the covariant Lie derivative.
See connection form for more details.

Nijenhuis–Lie derivative

Another generalization, due to Albert Nijenhuis, allows one to define the Lie derivative of a differential form along any section of the bundle Ω^k of differential forms with values in the tangent bundle. If K ∈ Ω^k and α is a differential p-form, then it is possible to define the interior product i_Kα of K and α. The Nijenhuis–Lie derivative is then the anticommutator of the interior product and the exterior derivative:

History

In 1931, Władysław Ślebodziński introduced a new differential operator, later called by David van Dantzig that of Lie derivation, which can be applied to scalars, vectors, tensors and affine connections and which proved to be a powerful instrument in the study of groups of automorphisms.
The Lie derivatives of general geometric objects were studied by A. Nijenhuis, Y. Tashiro and K. Yano.
For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians. In 1940, Léon Rosenfeld—and before him Wolfgang Pauli—introduced what he called a ‘local variation’ of a geometric object induced by an infinitesimal transformation of coordinates generated by a vector field. One can easily prove that his is.