Indifferential geometry, theLie derivative (/liː/LEE), named afterSophus Lie byWładysław Ślebodziński,^[1][2] evaluates the change of atensor field (including scalar functions,vector fields andone-forms), along theflow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on anydifferentiable manifold.

Functions, tensor fields and forms can be differentiated with respect to a vector field. IfT is a tensor field andX is a vector field, then the Lie derivative ofT with respect toX is denoted${\displaystyle {\mathcal{L}}_{X}T}$. Thedifferential operator${\displaystyle T\mapsto {\mathcal{L}}_{X}T}$ is aderivation of the algebra oftensor fields of the underlying manifold.

The Lie derivative commutes withcontraction and theexterior derivative ondifferential forms.

Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function orscalar 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 fieldY with respect to another vector fieldX is known as the "Lie bracket" ofX andY, and is often denoted [X,Y] instead of${\displaystyle {\mathcal{L}}_{X}Y}$. The space of vector fields forms aLie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensionalLie algebra representation of this Lie algebra, due to the identity

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

Proof of the identity

:${\displaystyle {\mathcal{L}}_{[X,Y]}T=[[X,Y],T]=[X,Y]T-T[X,Y]=([X,YT]-Y[X,T])-([X,TY]-[X,T]Y)=}$ ${\displaystyle =XYT-YTX-Y[X,T]-XTY+TYX+[X,T]Y=(XYT-YTX)+(TYX-XTY)-(Y[X,T]-[X,T]Y)=}$ ${\displaystyle =(X[Y,T]+[T,Y]X)-(Y[X,T]-[X,T]Y)=(X[Y,T]-[Y,T]X)-(Y[X,T]-[X,T]Y)=}$ ${\displaystyle =[X,[Y,T]]-[Y,[X,T]]={\mathcal{L}}_X{\mathcal{L}}_{Y}T-{\mathcal{L}}_Y{\mathcal{L}}_{X}T}$

valid for any vector fieldsX andY and any tensor fieldT.

Considering vector fields asinfinitesimal generators offlows (i.e. one-dimensionalgroups ofdiffeomorphisms) onM, the Lie derivative is thedifferential of the representation of thediffeomorphism group on tensor fields, analogous to Lie algebra representations asinfinitesimal representations associated togroup representation inLie group theory.

Generalisations exist forspinor fields,fibre bundles with aconnection andvector-valued differential forms.

A 'naïve' attempt to define the derivative of atensor field with respect to avector field would be to take thecomponents of the tensor field and take thedirectional derivative of each component with respect to the vector field. However, this definition is undesirable because it is not invariant underchanges of coordinate system, e.g. the naive derivative expressed inpolar orspherical coordinates differs from the naive derivative of the components inCartesian coordinates. On an abstractmanifold such a definition is meaningless and ill defined.

Indifferential geometry, there are three main coordinate independent notions of differentiation of tensor fields:

1. Lie derivatives,
2. derivatives with respect toconnections,
3. theexterior 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 atangent 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. aRiemannian metric in the case ofLevi-Civita connection, or just an abstractconnection) 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 fieldX at a pointp depends on the value ofX in a neighborhood ofp, not just atp 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${\displaystyle Y(p)}$ based at point${\displaystyle p}$ is transported by flowing its base point to${\displaystyle p^{\prime }}$, while flowing its tip point${\displaystyle p+Y(p)\delta }$ to${\displaystyle p^{\prime }+\delta p^{\prime }}$.

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.

Defining the derivative of a function${\displaystyle f:M\to \mathbb{R}}$ on a manifold is problematic because thedifference quotient${\displaystyle (f(x+h)-f(x))/h}$ cannot be determined while the displacement${\displaystyle x+h}$ is undefined.

The Lie derivative of a function${\displaystyle f:M\to \mathbb{R}}$ with respect to avector field${\displaystyle X}$ at a point${\displaystyle p\in M}$ is the function

${\displaystyle ({\mathcal{L}}_{X}f)(p)=\frac{d}{dt}{|}_{t=0}(f\circ {\mathrm{\Phi }}_X^t)(p)=\underset{t\to 0}{lim}\frac{f({\mathrm{\Phi }}_X^t(p))-f(p)}{t}}$

where${\displaystyle {\mathrm{\Phi }}_X^t(p)}$ is the point to which theflow defined by the vector field${\displaystyle X}$ maps the point${\displaystyle p}$ at time instant${\displaystyle t.}$ In the vicinity of${\displaystyle t=0,}$${\displaystyle {\mathrm{\Phi }}_X^t(p)}$ is the unique solution of the system

${\displaystyle \frac{d}{dt}{|}_t{\mathrm{\Phi }}_X^t(p)=X({\mathrm{\Phi }}_X^t(p))}$

of first-order autonomous (i.e. time-independent) differential equations, with${\displaystyle {\mathrm{\Phi }}_X^0(p)=p.}$

Setting${\displaystyle {\mathcal{L}}_{X}f={\mathrm{\nabla }}_{X}f}$ identifies the Lie derivative of a function with thedirectional derivative, which is also denoted by${\displaystyle X(f):={\mathcal{L}}_{X}f={\mathrm{\nabla }}_{X}f}$.

IfX andY are both vector fields, then the Lie derivative ofY with respect toX is also known as theLie bracket ofX andY, and is sometimes denoted${\displaystyle [X,Y]}$. 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 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${\displaystyle {\mathrm{\Phi }}_X^t:M\to M}$ be the corresponding local flow. Since${\displaystyle {\mathrm{\Phi }}_X^t}$ is a local diffeomorphism for each${\displaystyle t}$, it gives rise to apullback of tensor fields. For covariant tensors, this is just the multi-linear extension of thepullback map

$${\displaystyle {\left({\mathrm{\Phi }}_X^t\right)}_p^{\ast }:T_{{\mathrm{\Phi }}_X^t(p)}^{\ast }M\to T_p^{\ast }M,\left({\left({\mathrm{\Phi }}_X^t\right)}_p^{\ast }\alpha \right)(Y)=\alpha (T_p{\mathrm{\Phi }}_X^t(Y)),\alpha \in T_{{\mathrm{\Phi }}_X^t(p)}^{\ast }M,Y\in T_{p}M}$$For contravariant tensors, one extends the inverse

${\displaystyle {\left(T_p{\mathrm{\Phi }}_X^t\right)}^{-1}:T_{{\mathrm{\Phi }}_X^t(p)}M\to T_{p}M}$

of thedifferential${\displaystyle T_p{\mathrm{\Phi }}_X^t}$. For every${\displaystyle t,}$ there is, consequently, a tensor field${\displaystyle ({\mathrm{\Phi }}_X^t{)}^{\ast }T}$ 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${\displaystyle {\mathcal{L}}_{X}T}$ of${\displaystyle T}$ along a vector field${\displaystyle X}$ is defined at point${\displaystyle p\in M}$ to be

${\displaystyle {\mathcal{L}}_{X}T(p)=\frac{d}{dt}{|}_{t=0}{\left(({\mathrm{\Phi }}_X^t{)}^{\ast }T\right)}_p=\frac{d}{dt}{|}_{t=0}({\mathrm{\Phi }}_X^t{)}_p^{\ast }{T}_{{\mathrm{\Phi }}_X^t(p)}=\underset{t\to 0}{lim}\frac{({\mathrm{\Phi }}_X^t{)}^{\ast }{T}_{{\mathrm{\Phi }}_X^t(p)}-T_p}{t}.}$

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

More generally, for every smooth 1-parameter family${\displaystyle {\mathrm{\Phi }}_t}$ of diffeomorphisms that integrate a vector field${\displaystyle X}$ in the sense that${\displaystyle \frac{d}{dt}{|}_{t=0}{\mathrm{\Phi }}_t=X\circ {\mathrm{\Phi }}_0}$, one has$${\displaystyle {\mathcal{L}}_{X}T=({\mathrm{\Phi }}_0^{-1}{)}^{\ast }\frac{d}{dt}{|}_{t=0}{\mathrm{\Phi }}_t^{\ast }T=-\frac{d}{dt}{|}_{t=0}({\mathrm{\Phi }}_t^{-1}{)}^{\ast }{\mathrm{\Phi }}_0^{\ast }T.}$$

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 fieldsS andT, 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 tocontraction:

${\displaystyle {\mathcal{L}}_X(T(Y_1,\dots ,Y_n))=({\mathcal{L}}_{X}T)(Y_1,\dots ,Y_n)+T(({\mathcal{L}}_X{Y}_1),\dots ,Y_n)+\cdots +T(Y_1,\dots ,({\mathcal{L}}_X{Y}_n))}$

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 theLie bracket.

The Lie derivative acting on a differential form is theanticommutator of theinterior product with the exterior derivative. So if α is a differential form,

${\displaystyle {\mathcal{L}}_Y\alpha =i_{Y}d\alpha +d{i}_Y\alpha .}$

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. This isCartan's magic formula. Seeinterior product for details.

Explicitly, letT be a tensor field of type(p,q). ConsiderT to be a differentiablemultilinear map ofsmoothsectionsα¹,α², ...,α^p of the cotangent bundleT^∗M and of sectionsX₁,X₂, ...,X_q of thetangent bundleTM, writtenT(α¹,α², ...,X₁,X₂, ...) intoR. Define the Lie derivative ofT alongY by the formula

${\displaystyle ({\mathcal{L}}_{Y}T)({\alpha }_1,{\alpha }_2,\dots ,X_1,X_2,\dots )=Y(T({\alpha }_1,{\alpha }_2,\dots ,X_1,X_2,\dots ))}$

${\displaystyle -T({\mathcal{L}}_Y{\alpha }_1,{\alpha }_2,\dots ,X_1,X_2,\dots )-T({\alpha }_1,{\mathcal{L}}_Y{\alpha }_2,\dots ,X_1,X_2,\dots )-\dots }$

${\displaystyle -T({\alpha }_1,{\alpha }_2,\dots ,{\mathcal{L}}_Y{X}_1,X_2,\dots )-T({\alpha }_1,{\alpha }_2,\dots ,X_1,{\mathcal{L}}_Y{X}_2,\dots )-\dots }$

The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and theLeibniz rule for differentiation. The Lie derivative commutes with the contraction.

A particularly important class of tensor fields is the class ofdifferential forms. The restriction of the Lie derivative to the space of differential forms is closely related to theexterior 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 aninterior product, after which the relationships falls out as an identity known asCartan's formula. Cartan's formula can also be used as a definition of the Lie derivative on the space of differential forms.

LetM be a manifold andX a vector field onM. Let${\displaystyle \omega \in {\mathrm{\Lambda }}^k(M)}$ be ak-form, i.e., for each${\displaystyle p\in M}$,${\displaystyle \omega (p)}$ is analternatingmultilinear map from${\displaystyle (T_{p}M{)}^k}$ to the real numbers. Theinterior product ofX andω is the(k − 1)-form${\displaystyle i_X\omega }$ defined as

${\displaystyle (i_X\omega )(X_1,\dots ,X_{k-1})=\omega (X,X_1,\dots ,X_{k-1})}$

The differential form${\displaystyle i_X\omega }$ is also called thecontraction ofω withX, and

${\displaystyle i_X:{\mathrm{\Lambda }}^k(M)\to {\mathrm{\Lambda }}^{k-1}(M)}$

is a${\displaystyle \wedge }$-antiderivation where${\displaystyle \wedge }$ is thewedge product on differential forms. That is,${\displaystyle i_X}$ isR-linear, and

${\displaystyle i_X(\omega \wedge \eta )=(i_X\omega )\wedge \eta +(-1{)}^k\omega \wedge (i_X\eta )}$

for${\displaystyle \omega \in {\mathrm{\Lambda }}^k(M)}$ and η another differential form. Also, for a function${\displaystyle f\in {\mathrm{\Lambda }}^0(M)}$, that is, a real- or complex-valued function onM, one has

${\displaystyle i_{fX}\omega =f{i}_X\omega }$

where${\displaystyle fX}$ denotes the product off andX.The relationship betweenexterior derivatives and Lie derivatives can then be summarized as follows. First, since the Lie derivative of a functionf with respect to a vector fieldX is the same as the directional derivativeX(f), it is also the same as thecontraction of the exterior derivative off withX:

${\displaystyle {\mathcal{L}}_{X}f=i_{X}df}$

For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation inX:

${\displaystyle {\mathcal{L}}_X\omega =i_{X}d\omega +d(i_X\omega ).}$

This identity is known variously asCartan formula,Cartan homotopy formula orCartan's magic formula. Seeinterior 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

${\displaystyle d{\mathcal{L}}_X\omega ={\mathcal{L}}_X(d\omega ).}$

The Lie derivative also satisfies the relation

${\displaystyle {\mathcal{L}}_{fX}\omega =f{\mathcal{L}}_X\omega +df\wedge i_X\omega .}$

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

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

${\displaystyle ({\mathcal{L}}_{X}T{)}^{a_1\dots a_r}{}_{b_1\dots b_s}{\mathrm{\partial }}_{a_1}\otimes \cdots \otimes {\mathrm{\partial }}_{a_r}\otimes d{x}^{b_1}\otimes \cdots \otimes d{x}^{b_s}}$

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

The definition can be extended further totensor densities. IfT is a tensor density of some real number valued weightw (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 alinear connection${\displaystyle \mathrm{\Gamma }=({\mathrm{\Gamma }}_{bc}^a)}$, the Lie derivative along${\displaystyle X}$ is^[3]

${\displaystyle ({\mathcal{L}}_X\mathrm{\Gamma }{)}_{bc}^a=X^d{\mathrm{\partial }}_d{\mathrm{\Gamma }}_{bc}^a+{\mathrm{\partial }}_b{\mathrm{\partial }}_c{X}^a-{\mathrm{\Gamma }}_{bc}^d{\mathrm{\partial }}_d{X}^a+{\mathrm{\Gamma }}_{dc}^a{\mathrm{\partial }}_b{X}^d+{\mathrm{\Gamma }}_{bd}^a{\mathrm{\partial }}_c{X}^d}$

For clarity we now show the following examples in localcoordinate notation.

For ascalar field${\displaystyle \varphi (x^c)\in \mathcal{F}(M)}$ we have:

${\displaystyle ({\mathcal{L}}_X\varphi )=X(\varphi )=X^a{\mathrm{\partial }}_a\varphi }$.

Hence for the scalar field${\displaystyle \varphi (x,y)=x^2-\sin (y)}$ and the vector field${\displaystyle X^a{\mathrm{\partial }}_a=\sin (x){\mathrm{\partial }}_y-y^2{\mathrm{\partial }}_x}$ the corresponding Lie derivative becomes

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,

Some more abstract examples.

${\displaystyle {\mathcal{L}}_X(d{x}^b)=d{i}_X(d{x}^b)=d{X}^b={\mathrm{\partial }}_a{X}^{b}d{x}^a}$.

Hence for acovector field, i.e., adifferential form,${\displaystyle A=A_a(x^b)d{x}^a}$ we have:

${\displaystyle {\mathcal{L}}_{X}A=X(A_a)d{x}^a+A_b{\mathcal{L}}_X(d{x}^b)=(X^b{\mathrm{\partial }}_b{A}_a+A_b{\mathrm{\partial }}_a(X^b))d{x}^a}$

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

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

If${\displaystyle T=g}$ is the symmetric metric tensor, it is parallel with respect to theLevi-Civita connection (akacovariant derivative), and it becomes fruitful to use the connection. This has the effect of replacing all derivatives with covariant derivatives, giving

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

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

${\displaystyle {\mathcal{L}}_X:\mathcal{F}(M)\to \mathcal{F}(M)}$

is aderivation on the algebra${\displaystyle \mathcal{F}(M)}$. That is,${\displaystyle {\mathcal{L}}_X}$ isR-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 onM:^[4]

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

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 thetensor 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 theLie 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 theJacobi identity. Thus, one has the important result that the space of vector fields overM, equipped with the Lie bracket, forms aLie algebra.

The Lie derivative also has important properties when acting on differential forms. Letα andβ be two differential forms onM, and letX andY be two vector fields. Then

• ${\displaystyle {\mathcal{L}}_X(\alpha \wedge \beta )=({\mathcal{L}}_X\alpha )\wedge \beta +\alpha \wedge ({\mathcal{L}}_X\beta )}$
• ${\displaystyle [{\mathcal{L}}_X,{\mathcal{L}}_Y]\alpha :={\mathcal{L}}_X{\mathcal{L}}_Y\alpha -{\mathcal{L}}_Y{\mathcal{L}}_X\alpha ={\mathcal{L}}_{[X,Y]}\alpha }$
• ${\displaystyle [{\mathcal{L}}_X,i_Y]\alpha =[i_X,{\mathcal{L}}_Y]\alpha =i_{[X,Y]}\alpha ,}$ wherei denotes interior product defined above and it is clear whether [·,·] denotes thecommutator or theLie bracket of vector fields.

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

A definition for Lie derivatives ofspinors along generic spacetime vector fields, not necessarilyKilling ones, on a general (pseudo)Riemannian manifold was already proposed in 1971 byYvette Kosmann.^[5] Later, it was provided a geometric framework which justifies herad hoc prescription within the general framework of Lie derivatives onfiber bundles^[6] in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.^[7]

In a givenspin manifold, that is in a Riemannian manifold${\displaystyle (M,g)}$ admitting aspin structure, the Lie derivative of aspinorfield${\displaystyle \psi }$ can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via theAndré Lichnerowicz's local expression given in 1963:^[8]

${\displaystyle {\mathcal{L}}_X\psi :=X^a{\mathrm{\nabla }}_a\psi -\frac{1}{4}{\mathrm{\nabla }}_a{X}_b{\gamma }^a{\gamma }^b\psi ,}$

where${\displaystyle {\mathrm{\nabla }}_a{X}_b={\mathrm{\nabla }}_{[a}{X}_{b]}}$, as${\displaystyle X=X^a{\mathrm{\partial }}_a}$ is assumed to be aKilling vector field, and${\displaystyle {\gamma }^a}$ areDirac matrices.

It is then possible to extend Lichnerowicz's definition to all vector fields (generic infinitesimal transformations) by retaining Lichnerowicz's local expression for ageneric vector field${\displaystyle X}$, but explicitly taking the antisymmetric part of${\displaystyle {\mathrm{\nabla }}_a{X}_b}$ only.^[5] More explicitly, Kosmann's local expression given in 1972 is:^[5]

${\displaystyle {\mathcal{L}}_X\psi :=X^a{\mathrm{\nabla }}_a\psi -\frac{1}{8}{\mathrm{\nabla }}_{[a}{X}_{b]}[{\gamma }^a,{\gamma }^b]\psi ={\mathrm{\nabla }}_X\psi -\frac{1}{4}(d{X}^{\mathrm{♭}})\cdot \psi ,}$

where${\displaystyle [{\gamma }^a,{\gamma }^b]={\gamma }^a{\gamma }^b-{\gamma }^b{\gamma }^a}$ is the commutator,${\displaystyle d}$ isexterior derivative,${\displaystyle X^{\mathrm{♭}}=g(X,-)}$ is the dual 1 form corresponding to${\displaystyle X}$ under the metric (i.e. with lowered indices) and${\displaystyle \cdot }$ is Clifford multiplication.

It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of theconnection. 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 (covariant derivative), the dualisation of vector fields (lowering of the indices) and the Clifford multiplication on thespinor 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,^[9][10] 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 theKosmann 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 (such as the velocity vector${\displaystyle \overline{\psi }{\gamma }^a\psi }$ or the spin axial-vector${\displaystyle \overline{\psi }{\gamma }^a{\gamma }^5\psi }$) 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^[11].

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 (i.e. it has horizontal and vertical components), 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 fieldY overM (but not the principal bundle) but we also have aconnection over the principal bundle, we can define a vector field X over the principal bundle such that its horizontal component matchesY and its vertical component agrees with the connection. This is the covariant Lie derivative.

Seeconnection form for more details.

Another generalization, due toAlbert Nijenhuis, allows one to define the Lie derivative of a differential form along any section of the bundle Ω^k(M, TM) of differential forms with values in the tangent bundle. IfK ∈ Ω^k(M, TM) and α is a differentialp-form, then it is possible to define the interior producti_Kα ofK and α. The Nijenhuis–Lie derivative is then the anticommutator of the interior product and the exterior derivative:

${\displaystyle {\mathcal{L}}_K\alpha =[d,i_K]\alpha =d{i}_K\alpha -(-1{)}^{k-1}{i}_{K}d\alpha .}$

In 1931,Władysław Ślebodziński introduced a new differential operator, later called byDavid 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 ofnatural fiber bundles) were studied byA. Nijenhuis, Y. Tashiro andK. Yano.

For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians. In 1940,Léon Rosenfeld^[12]—and before him (in 1921)Wolfgang Pauli^[13]—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${\displaystyle -{\mathcal{L}}_X(A)}$.

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

• 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.

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

• Differential geometry
• Differential topology
• Differential operators
• Generalizations of the derivative

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