In differential geometry, the Lie derivative ( ), formulated by Władysław Ślebodziński and named after Sophus Lie, 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 . 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 (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:
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 (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 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 (i.e. time-independent) 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 (time-independent) 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 .
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 .
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:
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
Axiom 2. The Lie derivative obeys the following version of Leibniz's rule: For any tensor fields S and T, we have
Axiom 3. The Lie derivative obeys the Leibniz rule with respect to contraction:
Axiom 4. The Lie derivative commutes with exterior derivative on functions:
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 (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 . Consider T to be a differentiable multilinear map of smooth sections α1, α2, ..., αp of the cotangent bundle T∗M and of sections X1, X2, ..., Xq of the tangent bundle TM, written T(α1, α2, ..., X1, X2, ...) 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 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(f), 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 (e.g., the Levi Civita connection), then the partial derivative can be replaced with the covariant derivative which means replacing with (by abuse of notation) 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 (e.g. the volume density of weight 1), 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 (aka covariant derivative), and it becomes fruitful to use the connection. This has the effect of replacing all derivatives with covariant derivatives, giving
Properties