In differential geometry, the Lie derivative (/liː/ LEE), named after Sophus Lie by Władysław Ślebodziński,^[1][2] 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 ${\displaystyle {\mathcal {L}}_{X}T}$. The differential operator ${\displaystyle T\mapsto {\mathcal {L}}_{X}T}$ 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 ${\displaystyle {\mathcal {L}}_{X}Y}$. 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

${\displaystyle {\mathcal {L}}_{[X,Y]}T={\mathcal {L}}_{X}{\mathcal {L}}_{Y}T-{\mathcal {L}}_{Y}{\mathcal {L}}_{X}T,}$

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, and 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 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).

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 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 ${\displaystyle X}$ on a smooth manifold ${\displaystyle M,}$ let ${\displaystyle \Phi _{X}^{t}:M\to M}$ be the corresponding local flow. Since ${\displaystyle \Phi _{X}^{t}}$ is a local diffeomorphism for each ${\displaystyle t}$, it gives rise to a pullback of tensor fields. For covariant tensors, this is just the multi-linear extension of the pullback map

${\displaystyle \left(\Phi _{X}^{t}\right)_{p}^{*}:T_{\Phi _{X}^{t}(p)}^{*}M\to T_{p}^{*}M,\qquad \left(\Phi _{X}^{t}\right)_{p}^{*}\alpha (X)=\alpha {\bigl (}T_{p}\Phi _{X}^{t}(X){\bigr )},\quad \alpha \in T_{\Phi _{X}^{t}(p)}^{*}M,X\in T_{p}M}$

For contravariant tensors, one extends the inverse

${\displaystyle \left(T_{p}\Phi _{X}^{t}\right)^{-1}:T_{\Phi _{X}^{t}(p)}M\to T_{p}M}$

of the differential ${\displaystyle T_{p}\Phi _{X}^{t}}$. For every ${\displaystyle t,}$ there is, consequently, a tensor field ${\displaystyle (\Phi _{X}^{t})^{*}T}$ of the same type as ${\displaystyle T}$'s.

If ${\displaystyle T}$ is an ${\displaystyle (r,0)}$- or ${\displaystyle (0,s)}$-type tensor field, then the Lie derivative ${\displaystyle {\cal {L}}_{X}T}$ of ${\displaystyle T}$ along a vector field ${\displaystyle X}$ is defined at point ${\displaystyle p\in M}$ to be

${\displaystyle {\cal {L}}_{X}T(p)={\frac {d}{dt}}{\biggl |}_{t=0}\left({\bigl (}\Phi _{X}^{t}{\bigr )}^{*}T\right)_{p}={\frac {d}{dt}}{\biggl |}_{t=0}{\bigl (}\Phi _{X}^{t}{\bigr )}_{p}^{*}T_{\Phi _{X}^{t}(p)}=\lim _{t\to 0}{\frac {{\bigl (}\Phi _{X}^{t}{\bigr )}^{*}T_{\Phi _{X}^{t}(p)}-T_{p}}{t}}.}$

The resulting tensor field ${\displaystyle {\cal {L}}_{X}T}$ is of the same type as ${\displaystyle T}$'s.

More generally, for every smooth 1-parameter family ${\displaystyle \Phi _{t}}$of diffeomorphisms that integrate a vector field ${\displaystyle X}$ in the sense that ${\displaystyle {d \over dt}{\biggr |}_{t=0}\Phi _{t}=X\circ \Phi _{0}}$, one has

$${\displaystyle {\mathcal {L}}_{X}T={\bigl (}\Phi _{0}^{-1}{\bigr )}^{*}{d \over dt}{\biggr |}_{t=0}\Phi _{t}^{*}T=-{d \over dt}{\biggr |}_{t=0}{\bigl (}\Phi _{t}^{-1}{\bigr )}^{*}\Phi _{0}^{*}T\,.}$$

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

${\displaystyle {\mathcal {L}}_{Y}f=Y(f)}$

Axiom 2. The Lie derivative obeys the following version of Leibniz's rule: For any tensor fields S and T, we have

${\displaystyle {\mathcal {L}}_{Y}(S\otimes T)=({\mathcal {L}}_{Y}S)\otimes T+S\otimes ({\mathcal {L}}_{Y}T).}$

Axiom 3. The Lie derivative obeys the Leibniz rule with respect to contraction:

${\displaystyle {\mathcal {L}}_{X}(T(Y_{1},\ldots ,Y_{n}))=({\mathcal {L}}_{X}T)(Y_{1},\ldots ,Y_{n})+T(({\mathcal {L}}_{X}Y_{1}),\ldots ,Y_{n})+\cdots +T(Y_{1},\ldots ,({\mathcal {L}}_{X}Y_{n}))}$

Axiom 4. The Lie derivative commutes with exterior derivative on functions:

${\displaystyle [{\mathcal {L}}_{X},d]=0}$

If these axioms hold, then applying the Lie derivative ${\displaystyle {\mathcal {L}}_{X}}$ to the relation ${\displaystyle df(Y)=Y(f)}$ shows that

${\displaystyle {\mathcal {L}}_{X}Y(f)=X(Y(f))-Y(X(f)),}$

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,

${\displaystyle {\mathcal {L}}_{Y}\alpha =i_{Y}d\alpha +di_{Y}\alpha .}$

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.

Explicitly, let T be a tensor field of type (p, q). 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(α¹, α², ..., X₁, X₂, ...) into R. Define the Lie derivative of T along Y by the formula

${\displaystyle ({\mathcal {L}}_{Y}T)(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots )=Y(T(\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots ))}$

${\displaystyle -T({\mathcal {L}}_{Y}\alpha _{1},\alpha _{2},\ldots ,X_{1},X_{2},\ldots )-T(\alpha _{1},{\mathcal {L}}_{Y}\alpha _{2},\ldots ,X_{1},X_{2},\ldots )-\ldots }$

${\displaystyle -T(\alpha _{1},\alpha _{2},\ldots ,{\mathcal {L}}_{Y}X_{1},X_{2},\ldots )-T(\alpha _{1},\alpha _{2},\ldots ,X_{1},{\mathcal {L}}_{Y}X_{2},\ldots )-\ldots }$

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.

Coordinate expressions

In local coordinate notation, for a type (r, s) tensor field ${\displaystyle T}$, the Lie derivative along ${\displaystyle X}$ is

${\displaystyle {\begin{aligned}({\mathcal {L}}_{X}T)^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}={}&X^{c}(\partial _{c}T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}})\\&{}-{}(\partial _{c}X^{a_{1}})T^{ca_{2}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}-\ldots -(\partial _{c}X^{a_{r}})T^{a_{1}\ldots a_{r-1}c}{}_{b_{1}\ldots b_{s}}\\&+(\partial _{b_{1}}X^{c})T^{a_{1}\ldots a_{r}}{}_{cb_{2}\ldots b_{s}}+\ldots +(\partial _{b_{s}}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s-1}c}\end{aligned}}}$

here, the notation ${\displaystyle \partial _{a}={\frac {\partial }{\partial x^{a}}}}$ means taking the partial derivative with respect to the coordinate ${\displaystyle x^{a}}$. Alternatively, if we are using a torsion-free connection (e.g., the Levi Civita connection), then the partial derivative ${\displaystyle \partial _{a}}$ can be replaced with the covariant derivative which means replacing ${\displaystyle \partial _{a}X^{b}}$ with (by abuse of notation) ${\displaystyle \nabla _{a}X^{b}=X_{;a}^{b}:=(\nabla X)_{a}^{\ b}=\partial _{a}X^{b}+\Gamma _{ac}^{b}X^{c}}$ where the ${\displaystyle \Gamma _{bc}^{a}=\Gamma _{cb}^{a}}$ 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

${\displaystyle ({\mathcal {L}}_{X}T)^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}\partial _{a_{1}}\otimes \cdots \otimes \partial _{a_{r}}\otimes dx^{b_{1}}\otimes \cdots \otimes dx^{b_{s}}}$

which is independent of any coordinate system and of the same type as ${\displaystyle T}$.

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.

${\displaystyle {\begin{aligned}({\mathcal {L}}_{X}T)^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}={}&X^{c}(\partial _{c}T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}})-(\partial _{c}X^{a_{1}})T^{ca_{2}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}-\ldots -(\partial _{c}X^{a_{r}})T^{a_{1}\ldots a_{r-1}c}{}_{b_{1}\ldots b_{s}}+\\&+(\partial _{b_{1}}X^{c})T^{a_{1}\ldots a_{r}}{}_{cb_{2}\ldots b_{s}}+\ldots +(\partial _{b_{s}}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s-1}c}+w(\partial _{c}X^{c})T^{a_{1}\ldots a_{r}}{}_{b_{1}\ldots b_{s}}\end{aligned}}}$

Notice the new term at the end of the expression.

For a linear connection ${\displaystyle \Gamma =(\Gamma _{bc}^{a})}$, the Lie derivative along ${\displaystyle X}$ is^[3]

${\displaystyle ({\mathcal {L}}_{X}\Gamma )_{bc}^{a}=X^{d}\partial _{d}\Gamma _{bc}^{a}+\partial _{b}\partial _{c}X^{a}-\Gamma _{bc}^{d}\partial _{d}X^{a}+\Gamma _{dc}^{a}\partial _{b}X^{d}+\Gamma _{bd}^{a}\partial _{c}X^{d}}$

Examples

For clarity we now show the following examples in local coordinate notation.

For a scalar field ${\displaystyle \phi (x^{c})\in {\mathcal {F}}(M)}$ we have:

${\displaystyle ({\mathcal {L}}_{X}\phi )=X(\phi )=X^{a}\partial _{a}\phi }$.

Hence for the scalar field ${\displaystyle \phi (x,y)=x^{2}-\sin(y)}$ and the vector field ${\displaystyle X=\sin(x)\partial _{y}-y^{2}\partial _{x}}$ the corresponding Lie derivative becomes

$${\displaystyle {\begin{alignedat}{3}{\mathcal {L}}_{X}\phi &=(\sin(x)\partial _{y}-y^{2}\partial _{x})(x^{2}-\sin(y))\\&=\sin(x)\partial _{y}(x^{2}-\sin(y))-y^{2}\partial _{x}(x^{2}-\sin(y))\\&=-\sin(x)\cos(y)-2xy^{2}\\\end{alignedat}}}$$

For an example of higher rank differential form, consider the 2-form ${\displaystyle \omega =(x^{2}+y^{2})dx\wedge dz}$ and the vector field ${\displaystyle X}$ from the previous example. Then,

$${\displaystyle {\begin{aligned}{\mathcal {L}}_{X}\omega &=d(i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}((x^{2}+y^{2})dx\wedge dz))+i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(d((x^{2}+y^{2})dx\wedge dz))\\&=d(-y^{2}(x^{2}+y^{2})dz)+i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(2ydy\wedge dx\wedge dz)\\&=\left(-2xy^{2}dx+(-2yx^{2}-4y^{3})dy\right)\wedge dz+(2y\sin(x)dx\wedge dz+2y^{3}dy\wedge dz)\\&=\left(-2xy^{2}+2y\sin(x)\right)dx\wedge dz+(-2yx^{2}-2y^{3})dy\wedge dz\end{aligned}}}$$

Some more abstract examples.

${\displaystyle {\mathcal {L}}_{X}(dx^{b})=di_{X}(dx^{b})=dX^{b}=\partial _{a}X^{b}dx^{a}}$.

Hence for a covector field, i.e., a differential form, ${\displaystyle A=A_{a}(x^{b})dx^{a}}$ we have:

${\displaystyle {\mathcal {L}}_{X}A=X(A_{a})dx^{a}+A_{b}{\mathcal {L}}_{X}(dx^{b})=(X^{b}\partial _{b}A_{a}+A_{b}\partial _{a}(X^{b}))dx^{a}}$

The coefficient of the last expression is the local coordinate expression of the Lie derivative.

For a covariant rank 2 tensor field ${\displaystyle T=T_{ab}(x^{c})dx^{a}\otimes dx^{b}}$ we have:

$${\displaystyle {\begin{aligned}({\mathcal {L}}_{X}T)&=({\mathcal {L}}_{X}T)_{ab}dx^{a}\otimes dx^{b}\\&=X(T_{ab})dx^{a}\otimes dx^{b}+T_{cb}{\mathcal {L}}_{X}(dx^{c})\otimes dx^{b}+T_{ac}dx^{a}\otimes {\mathcal {L}}_{X}(dx^{c})\\&=(X^{c}\partial _{c}T_{ab}+T_{cb}\partial _{a}X^{c}+T_{ac}\partial _{b}X^{c})dx^{a}\otimes dx^{b}\\\end{aligned}}}$$

If ${\displaystyle T=g}$ 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

${\displaystyle ({\mathcal {L}}_{X}g)=(X^{c}g_{ab;c}+g_{cb}X_{;a}^{c}+g_{ac}X_{;b}^{c})dx^{a}\otimes dx^{b}=(X_{b;a}+X_{a;b})dx^{a}\otimes dx^{b}}$

Properties

The Lie derivative has a number of properties. Let ${\displaystyle {\mathcal {F}}(M)}$ be the algebra of functions defined on the manifold M. Then

${\displaystyle {\mathcal {L}}_{X}:{\mathcal {F}}(M)\rightarrow {\mathcal {F}}(M)}$

is a derivation on the algebra ${\displaystyle {\mathcal {F}}(M)}$. That is, ${\displaystyle {\mathcal {L}}_{X}}$ is R-linear and

${\displaystyle {\mathcal {L}}_{X}(fg)=({\mathcal {L}}_{X}f)g+f{\mathcal {L}}_{X}g.}$

Similarly, it is a derivation on ${\displaystyle {\mathcal {F}}(M)\times {\mathcal {X}}(M)}$ where ${\displaystyle {\mathcal {X}}(M)}$ is the set of vector fields on M (cf. Theorem 6 from the article: Nichita, F.F. Unification Theories: New Results and Examples. Axioms 2019, 8, 60):

${\displaystyle {\mathcal {L}}_{X}(fY)=({\mathcal {L}}_{X}f)Y+f{\mathcal {L}}_{X}Y}$

which may also be written in the equivalent notation

${\displaystyle {\mathcal {L}}_{X}(f\otimes Y)=({\mathcal {L}}_{X}f)\otimes Y+f\otimes {\mathcal {L}}_{X}Y}$

where the tensor product symbol ${\displaystyle \otimes }$ 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,

${\displaystyle {\mathcal {L}}_{X}[Y,Z]=[{\mathcal {L}}_{X}Y,Z]+[Y,{\mathcal {L}}_{X}Z]}$

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

• ${\displaystyle {\mathcal {L}}_{X}(\alpha \wedge \beta )=({\mathcal {L}}_{X}\alpha )\wedge \beta +\alpha \wedge ({\mathcal {L}}_{X}\beta )}$
• ${\displaystyle [{\mathcal {L}}_{X},{\mathcal {L}}_{Y}]\alpha$ :={\mathcal {L}}_{X}{\mathcal {L}}_{Y}\alpha -{\mathcal {L}}_{Y}{\mathcal {L}}_{X}\alpha ={\mathcal {L}}_{[X,Y]}\alpha }
• ${\displaystyle [{\mathcal {L}}_{X},i_{Y}]\alpha =[i_{X},{\mathcal {L}}_{Y}]\alpha =i_{[X,Y]}\alpha ,}$ 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.

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 (i.e., sections of natural fiber bundles) 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^[10]—and before him (in 1921^[11]) Wolfgang Pauli^[12]—introduced what he called a ‘local variation’ ${\displaystyle \delta ^{\ast }A}$ of a geometric object ${\displaystyle A\,}$ induced by an infinitesimal transformation of coordinates generated by a vector field ${\displaystyle X\,}$. One can easily prove that his ${\displaystyle \delta ^{\ast }A}$ is ${\displaystyle -{\mathcal {L}}_{X}(A)\,}$.

See also

• Covariant derivative
• Connection (mathematics)
• Frölicher–Nijenhuis bracket
• Geodesic
• Killing field
• Derivative of the exponential map

Notes

References

• Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. London: Benjamin-Cummings. ISBN 0-8053-0102-X. See section 2.2.
• Bleecker, David (1981). Gauge Theory and Variational Principles. Addison-Wesley. ISBN 0-201-10096-7. See Chapter 0.
• Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Berlin: Springer. ISBN 3-540-42627-2. See section 1.6.
• Kolář, I.; Michor, P.; Slovák, J. (1993). Natural operations in differential geometry. Springer-Verlag. ISBN 9783662029503. Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
• Lang, S. (1995). Differential and Riemannian manifolds. Springer-Verlag. ISBN 978-0-387-94338-1. For generalizations to infinite dimensions.
• Lang, S. (1999). Fundamentals of Differential Geometry. Springer-Verlag. ISBN 978-0-387-98593-0. For generalizations to infinite dimensions.
• Yano, K. (1957). The Theory of Lie Derivatives and its Applications. North-Holland. ISBN 978-0-7204-2104-0. Classical approach using coordinates.

External links

• "Lie derivative", Encyclopedia of Mathematics, EMS Press, 2001 [1994]