Lie derivative

 ( 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, and ), 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 $\backslash mathcal\{L\}\_X\; T$. The differential operator $T\; \backslash mapsto\; \backslash mathcal\{L\}\_X\; T$ 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 $\backslash mathcal\{L\}\_X\; Y$. 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

$\backslash mathcal\{L\}\_\{X,Y\}\; T\; =\; \backslash mathcal\{L\}\_X\; \backslash mathcal\{L\}\_\{Y\}\; T\; -\; \backslash mathcal\{L\}\_Y\; \backslash mathcal\{L\}\_X\; T,$

Proof of the identity

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

valid for any vector fields X and Y and any tensor field T.

Considering vector fields as Lie algebra of flows (i.e. one-dimensional groups of ) 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, with a connection and vector-valued differential forms.

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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:

1. Lie derivatives,
2. derivatives with respect to connections,
3. 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 space 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 $Y(p)$ based at point $p$ is transported by flowing its base point to $p\text{'}$, while flowing its tip point $p\; +\; Y(p)\; \backslash delta$ to $p\text{'}\; +\; \backslash delta\; p\text{'}$.

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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.

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The (Lie) derivative of a function Defining the derivative of a function $f\backslash colon\; M\; \backslash to\; \{\backslash mathbb\; R\}$ on a manifold takes care because the difference quotient $\backslash textstyle\; (f(x+h)-f(x))/h$ cannot be determined while the displacement $x+h$ is undefined.

The Lie derivative of a function $f\backslash colon\; M\backslash to\; \{\backslash mathbb\; R\}$ with respect to a vector field $X$ at a point $p\; \backslash in\; M$ is the function

$(\backslash mathcal\{L\}\_X\; f)\; (p)\; =\; \{d\; \backslash over\; dt\}\; \backslash biggr|\_\{t=0\}\; \backslash bigl(f\; \backslash circ\; \backslash Phi^t\_X\backslash bigr)(p)\; =\; \backslash lim\_\{t\backslash to\; 0\}\; \backslash frac\{f\backslash bigl(\backslash Phi^t\_X(p)\backslash bigr)\; -\; f\backslash bigl(p\backslash bigr)\}\{t\}$

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

$$

\frac{d}{dt}\biggr|_t \Phi^t_X(p) = X\bigl(\Phi^t_X(p)\bigr) of first-order autonomous (i.e. time-independent) differential equations, with $\backslash Phi^0\_X(p)\; =\; p.$

Setting $\backslash mathcal\{L\}\_X\; f\; =\; \backslash nabla\_X\; f$ identifies the Lie derivative of a function with the directional derivative, which is also denoted by $X(f):=\; \backslash mathcal\{L\}\_X\; f\; =\; \backslash nabla\_X\; f$.

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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 $X,Y$. 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:

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The Lie derivative of a tensor field

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

$$\backslash left(\backslash Phi^t\_X\backslash right)^*\_p\; :\; T^*\_\{\backslash Phi^t\_X(p)\}M\; \backslash to\; T^*\_\{p\}M,\; \backslash qquad\; \backslash left(\backslash left(\backslash Phi^t\_X\backslash right)^*\_p\; \backslash alpha\backslash right)\; (Y)\; =\; \backslash alpha\backslash bigl(T\_p\; \backslash Phi^t\_X(Y)\backslash bigr),\; \backslash quad\; \backslash alpha\; \backslash in\; T^*\_\{\backslash Phi^t\_X(p)\}M,\; Y\; \backslash in\; T\_\{p\}M$$ For contravariant tensors, one extends the inverse

$\backslash left(T\_p\backslash Phi^t\_X\backslash right)^\{-1\}\; :\; T\_\{\backslash Phi^t\_X(p)\}M\; \backslash to\; T\_\{p\}M$

of the differential $T\_p\backslash Phi^t\_X$. For every $t,$ there is, consequently, a tensor field $(\backslash Phi^t\_X)^*\; T$ of the same type as $T$'s.

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

$\{\backslash cal\; L\}\_X\; T(p)\; =\; \backslash frac\{d\}\{dt\}\backslash biggl|\_\{t=0\}\; \backslash left(\backslash bigl(\backslash Phi^t\_X\backslash bigr)^*\; T\backslash right)\_p\; =\; \backslash frac\{d\}\{dt\}\backslash biggl|\_\{t=0\}\backslash bigl(\backslash Phi^t\_X\backslash bigr)^*\_p\; T\_\{\backslash Phi^t\_X(p)\}$

= \lim_{t \to 0}\frac{\bigl(\Phi^t_X\bigr)^*T_{\Phi^t_X(p)} - T_p}{t}.

The resulting tensor field $\{\backslash cal\; L\}\_X\; T$ is of the same type as $T$'s.

More generally, for every smooth 1-parameter family $\backslash Phi\_t$ of diffeomorphisms that integrate a vector field $X$ in the sense that $\{d\; \backslash over\; dt\}\backslash biggr|\_\{t=0\}\; \backslash Phi\_t\; =\; X\; \backslash circ\; \backslash Phi\_0$, one has$$\backslash mathcal\{L\}\_X\; T\; =\; \backslash bigl(\backslash Phi\_0^\{-1\}\backslash bigr)^*\; \{d\; \backslash over\; dt\}\backslash biggr|\_\{t=0\}\; \backslash Phi\_t^*\; T\; =\; -\; \{d\; \backslash over\; dt\}\backslash biggr|\_\{t=0\}\; \backslash bigl(\backslash Phi\_t^\{-1\}\backslash bigr)^*\; \backslash Phi\_0^*\; T\; \backslash ,\; .$$

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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

:$\backslash mathcal\{L\}\_Yf=Y(f)$

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

:$\backslash mathcal\{L\}\_Y(S\backslash otimes\; T)=(\backslash mathcal\{L\}\_YS)\backslash otimes\; T+S\backslash otimes\; (\backslash mathcal\{L\}\_YT).$

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

:$\backslash mathcal\{L\}\_X\; (T(Y\_1,\; \backslash ldots,\; Y\_n))\; =\; (\backslash mathcal\{L\}\_X\; T)(Y\_1,\backslash ldots,\; Y\_n)\; +\; T((\backslash mathcal\{L\}\_X\; Y\_1),\; \backslash ldots,\; Y\_n)\; +\; \backslash cdots\; +\; T(Y\_1,\; \backslash ldots,\; (\backslash mathcal\{L\}\_X\; Y\_n))$

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

:$\backslash mathcal\{L\}\_X,\; =\; 0$

Using the first and third axioms, applying the Lie derivative $\backslash mathcal\{L\}\_X$ to $Y(f)$ shows that

:$\backslash mathcal\{L\}\_X\; Y\; (f)\; =\; X(Y(f))\; -\; Y(X(f)),$

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,

:$\backslash mathcal\{L\}\_Y\backslash alpha=i\_Yd\backslash alpha+di\_Y\backslash alpha.$

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 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 function 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

$(\backslash mathcal\{L\}\_Y\; T)(\backslash alpha\_1,\; \backslash alpha\_2,\; \backslash ldots,\; X\_1,\; X\_2,\; \backslash ldots)\; =Y(T(\backslash alpha\_1,\backslash alpha\_2,\backslash ldots,X\_1,X\_2,\backslash ldots))$

:$-\; T(\backslash mathcal\{L\}\_Y\backslash alpha\_1,\; \backslash alpha\_2,\; \backslash ldots,\; X\_1,\; X\_2,\; \backslash ldots)$

- T(\alpha_1, \mathcal{L}_Y\alpha_2, \ldots, X_1, X_2, \ldots) -\ldots

:$-\; T(\backslash alpha\_1,\; \backslash alpha\_2,\; \backslash ldots,\; \backslash mathcal\{L\}\_YX\_1,\; X\_2,\; \backslash ldots)$

- T(\alpha_1, \alpha_2, \ldots, X_1, \mathcal{L}_YX_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.

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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 $\backslash omega\; \backslash in\; \backslash Lambda^k(M)$ be a k-form, i.e., for each $p\; \backslash in\; M$, $\backslash omega(p)$ is an Alternating form multilinear map from $(T\_p\; M)^k$ to the real numbers. The interior product of X and ω is the -form $i\_X\backslash omega$ defined as

$(i\_X\backslash omega)\; (X\_1,\; \backslash ldots,\; X\_\{k-1\})\; =\; \backslash omega\; (X,X\_1,\; \backslash ldots,\; X\_\{k-1\})\backslash ,$

The differential form $i\_X\backslash omega$ is also called the contraction of ω with X, and

$i\_X:\backslash Lambda^k(M)\; \backslash rightarrow\; \backslash Lambda^\{k-1\}(M)$

is a $\backslash wedge$-antiderivation where $\backslash wedge$ is the wedge product on differential forms. That is, $i\_X$ is R-linear, and

$i\_X\; (\backslash omega\; \backslash wedge\; \backslash eta)\; =\; (i\_X\; \backslash omega)\; \backslash wedge\; \backslash eta\; +\; (-1)^k\; \backslash omega\; \backslash wedge\; (i\_X\; \backslash eta)$

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

$i\_\{fX\}\; \backslash omega\; =\; f\backslash ,i\_X\backslash omega$

where $f\; X$ 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:

$\backslash mathcal\{L\}\_Xf\; =\; i\_X\; \backslash ,\; df$

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

$\backslash mathcal\{L\}\_X\backslash omega\; =\; i\_Xd\backslash omega\; +\; d(i\_X\; \backslash omega).$

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

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

The Lie derivative also satisfies the relation

$\backslash mathcal\{L\}\_\{fX\}\backslash omega\; =\; f\backslash mathcal\{L\}\_X\backslash omega\; +\; df\; \backslash wedge\; i\_X\; \backslash omega\; .$

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Coordinate expressions In local coordinate notation, for a type tensor field $T$, the Lie derivative along $X$ is

$\backslash begin\{align\}$

 (\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 ^{c a_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}{}_{c b_2 \ldots b_s} + \ldots + (\partial_{b_s}X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c}
     

\end{align}

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

$(\backslash mathcal\{L\}\_X\; T)\; ^\{a\_1\; \backslash ldots\; a\_r\}\{\}\_\{b\_1\; \backslash ldots\; b\_s\}\backslash partial\_\{a\_1\}\backslash otimes\backslash cdots\backslash otimes\backslash partial\_\{a\_r\}\backslash otimes\; dx^\{b\_1\}\backslash otimes\backslash cdots\backslash otimes\; dx^\{b\_s\}$

which is independent of any coordinate system and of the same type as $T$.

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.

$\backslash begin\{align\}$

 (\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 ^{c a_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}{}_{c b_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{align}

Notice the new term at the end of the expression.

For a linear connection $\backslash Gamma\; =\; (\; \backslash Gamma^\{a\}\_\{bc\}\; )$, the Lie derivative along $X$ is

$$

(\mathcal{L}_X \Gamma)^{a}_{bc} = X^d\partial_d \Gamma^{a}_{bc} + \partial_b\partial_c X^a - \Gamma^{d}_{bc}\partial_d X^a + \Gamma^{a}_{dc}\partial_b X^d + \Gamma^{a}_{bd}\partial_c X^d

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

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

$(\backslash mathcal\; \{L\}\_X\; \backslash phi)\; =\; X(\backslash phi)\; =\; X^a\; \backslash partial\_a\; \backslash phi$.

Hence for the scalar field $\backslash phi(x,y)\; =\; x^2\; -\; \backslash sin(y)$ and the vector field $X^a\backslash partial\_a\; =\; \backslash sin(x)\backslash partial\_y\; -\; y^2\backslash partial\_x$ the corresponding Lie derivative becomes

                 & = \sin(x)\partial_y(x^2 - \sin(y)) - y^2\partial_x(x^2 - \sin(y))\\
                 & = -\sin(x)\cos(y) - 2xy^2 \\
     

\end{alignat}

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

Some more abstract examples.

$\backslash mathcal\{L\}\_X\; (dx^b)\; =\; d\; i\_X\; (dx^b)\; =\; d\; X^b\; =\; \backslash partial\_a\; X^b\; dx^a$.

Hence for a one-form, i.e., a differential form, $A\; =\; A\_a(x^b)dx^a$ we have:

$\backslash mathcal\{L\}\_X\; A\; =\; X\; (A\_a)\; dx^a\; +\; A\_b\; \backslash mathcal\{L\}\_X\; (dx^b)\; =\; (X^b\; \backslash partial\_b\; A\_a\; +\; A\_b\backslash partial\_a\; (X^b))dx^a$

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

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

                  &= 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{align}

If $T\; =\; g$ 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

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

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

$\backslash mathcal\{L\}\_X\; :\; \backslash mathcal\{F\}(M)\; \backslash rightarrow\; \backslash mathcal\{F\}(M)$

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

$\backslash mathcal\{L\}\_X(fg)\; =\; (\backslash mathcal\{L\}\_Xf)\; g\; +\; f\backslash mathcal\{L\}\_Xg.$

Similarly, it is a derivation on $\backslash mathcal\{F\}(M)\; \backslash times\; \backslash mathcal\{X\}(M)$ where $\backslash mathcal\{X\}(M)$ is the set of vector fields on M:

$\backslash mathcal\{L\}\_X(fY)\; =\; (\backslash mathcal\{L\}\_Xf)\; Y\; +\; f\backslash mathcal\{L\}\_X\; Y$

which may also be written in the equivalent notation

$\backslash mathcal\{L\}\_X(f\backslash otimes\; Y)\; =\; (\backslash mathcal\{L\}\_Xf)\; \backslash otimes\; Y\; +\; f\backslash otimes\; \backslash mathcal\{L\}\_X\; Y$

where the tensor product symbol $\backslash otimes$ 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,

$\backslash mathcal\{L\}\_X\; Y,Z\; =\; \backslash mathcal\{L\}\_X\; +\; Y,\backslash mathcal\{L\}\_X$

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

• $\backslash mathcal\{L\}\_X(\backslash alpha\backslash wedge\backslash beta)\; =\; (\backslash mathcal\{L\}\_X\backslash alpha)\; \backslash wedge\backslash beta\; +\; \backslash alpha\backslash wedge\; (\backslash mathcal\{L\}\_X\backslash beta)$
• $\backslash mathcal\{L\}\_X,\backslash mathcal\{L\}\_Y\backslash alpha\; :=\; \backslash mathcal\{L\}\_X\backslash mathcal\{L\}\_Y\backslash alpha-\backslash mathcal\{L\}\_Y\backslash mathcal\{L\}\_X\backslash alpha\; =\; \backslash mathcal\{L\}\_\{X,Y\}\backslash alpha$
• $\backslash mathcal\{L\}\_X,i\_Y\backslash alpha\; =\; i\_X,\backslash mathcal\{L\}\_Y\backslash alpha\; =\; i\_\{X,Y\}\backslash alpha,$ where i denotes interior product defined above and it is clear whether ·,· denotes the commutator or the Lie bracket of vector fields.

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

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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 (pseudo) 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

A. Trautman (1972). 9780198511267, Clarenden Press. ISBN 9780198511267

in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.

In a given spin manifold, that is in a Riemannian manifold $(M,g)$ admitting a spin structure, the Lie derivative of a spinor field $\backslash psi$ can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via the André Lichnerowicz's local expression given in 1963:

$\backslash mathcal\{L\}\_X\; \backslash psi\; :=\; X^\{a\}\backslash nabla\_\{a\}\backslash psi\; -\; \backslash frac14\backslash nabla\_\{a\}X\_\{b\}\; \backslash gamma^\{a\}\backslash gamma^\{b\}\backslash psi\backslash ,\; ,$

where $\backslash nabla\_\{a\}X\_\{b\}\; =\; \backslash nabla\_\{a\}X\_\{b\}$, as $X\; =\; X^\{a\}\backslash partial\_\{a\}$ is assumed to be a Killing vector field, and $\backslash gamma^\{a\}$ are Dirac matrices.

It is then possible to extend Lichnerowicz's definition to all vector fields (generic infinitesimal transformations) by retaining Lichnerowicz's local expression for a generic vector field $X$, but explicitly taking the antisymmetric part of $\backslash nabla\_\{a\}X\_\{b\}$ only. More explicitly, Kosmann's local expression given in 1972 is:

$\backslash mathcal\{L\}\_X\; \backslash psi\; :=\; X^\{a\}\backslash nabla\_\{a\}\backslash psi\; -\; \backslash frac18\backslash nabla\_\{a\}X\_\{b\}\backslash gamma^\{a\},\backslash gamma^\{b\}\backslash psi\backslash ,\; =\; \backslash nabla\_X\; \backslash psi\; -\; \backslash frac14\; (d\; X^\backslash flat)\backslash cdot\; \backslash psi\backslash ,\; ,$

where $\backslash gamma^\{a\},\backslash gamma^\{b\}=\; \backslash gamma^a\backslash gamma^b\; -\; \backslash gamma^b\backslash gamma^a$ is the commutator, $d$ is exterior derivative, $X^\backslash flat\; =\; g(X,\; -)$ is the dual 1 form corresponding to $X$ under the metric (i.e. with lowered indices) and $\backslash cdot$ 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 (covariant derivative), the dualisation of vector fields (lowering of the indices) 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 (such as the velocity vector $\backslash overline\{\backslash psi\}\backslash gamma^\{a\}\backslash psi$ or the spin axial-vector $\backslash overline\{\backslash psi\}\backslash gamma^\{a\}\backslash gamma^5\backslash psi$) 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.

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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 (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 field Y over M (but not the principal bundle) 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.

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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( M, T M) of differential forms with values in the tangent bundle. If K ∈ Ω ^k( M, T M) 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:

$\backslash mathcal\{L\}\_K\backslash alpha=d,i\_K\backslash alpha\; =\; di\_K\backslash alpha-(-1)^\{k-1\}i\_K\; \backslash ,\; d\backslash alpha.$

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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 bundle) were studied by Albert 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 (in 1921) Wolfgang Pauli

See section 23—introduced what he called a ‘local variation’ $\backslash delta^\{\backslash ast\}A$ of a geometric object $A\backslash ,$ induced by an infinitesimal transformation of coordinates generated by a vector field $X\backslash ,$. One can easily prove that his $\backslash delta^\{\backslash ast\}A$ is $-\; \backslash mathcal\{L\}\_X(A)\backslash ,$.

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See also

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

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Notes

• (1978). 080530102X, Benjamin-Cummings. 080530102X
  See section 2.2.
• David Bleecker (1981). 9780201100969, Addison-Wesley. . ISBN 9780201100969
  See Chapter 0.
• Jürgen Jost (2025). 9783540426271, Springer. ISBN 9783540426271
  See section 1.6.
• (1993). 9783662029503, Springer-Verlag. . ISBN 9783662029503
  Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
• S. Lang (1995). 9780387943381, Springer-Verlag. ISBN 9780387943381
  For generalizations to infinite dimensions.
• S. Lang (1999). 9780387985930, Springer-Verlag. ISBN 9780387985930
  For generalizations to infinite dimensions.
• K. Yano (2025). 9780720421040, North-Holland. . ISBN 9780720421040
  Classical approach using coordinates.

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External links

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