Lie differentiation

A natural operation on a differentiable manifold $M$ that associates with a differentiable vector field $X$ and a differentiable geometric object $Q$ on $M$( cf. Geometric objects, theory of) a new geometric object ${\mathcal L} _ {X} Q$, which describes the rate of change of $Q$ with respect to the one-parameter (local) transformation group $\phi _ {t}$ of $M$ generated by $X$. The geometric object ${\mathcal L} _ {X} Q$ is called the Lie derivative of the geometric object $Q$ with respect to $X$( cf. also Lie derivative). Here it is assumed that transformations of $M$ induce transformations in the space of objects $Q$ in a natural way.

In the special case when $Q$ is a vector-valued function on $M$, its Lie derivative ${\mathcal L} _ {X} Q$ coincides with the derivative $\partial _ {X} Q$ of the function $Q$ in the direction of the vector field $X$ and is given by the formula

$$\left . ( {\mathcal L} _ {X} Q ) ( x) = \ \frac{d}{dt} Q \circ \phi _ {t} ( x) \right | _ {t=} 0 ,\ x \in M ,$$

where $\phi _ {t}$ is the one-parameter local transformation group on $M$ generated by $X$, or, in the local coordinates $x ^ {i}$, by the formula

$${\mathcal L} _ {X} Q ( x ^ {i} ) = \ \sum _ { j } X ^ {j} \frac \partial {\partial x ^ {j} } Q ( x ^ {i} ) ,$$

where

$$X = \sum _ { j } X ^ {j} ( x) \frac \partial {\partial x ^ {j} } .$$

In the general case the definition of Lie differentiation consists in the following. Let $W$ be a $\mathop{\rm GL} ^ {k} ( n)$- space, that is, a manifold with a fixed action of the general differential group $\mathop{\rm GL} ^ {k} ( n)$ of order $k$( the group of $k$- jets at the origin of diffeomorphisms $\phi : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n}$, $\phi ( 0) = 0$). Let $Q : P ^ {k} M \rightarrow W$ be a geometric object of order $k$ and type $W$ on an $n$- dimensional manifold $M$, regarded as a $\mathop{\rm GL} ^ {k} ( n)$- equivariant mapping of the principal $\mathop{\rm GL} ^ {k} ( n)$- bundle of coframes $P ^ {k} M$ of order $k$ on $M$ into $W$. The one-parameter local transformation group $\phi _ {t}$ on $M$ generated by a vector field $X$ on $M$ induces a one-parameter local transformation group $\phi _ {t} ^ {(} k)$ on the manifold of coframes $P ^ {k} M$. Its velocity field

$$X ^ {(} k) = \left . \frac{d}{dt} \phi _ {t} ^ {(} k) \right | _ {t=} 0$$

is called the complete lift of $X$ to $P ^ {k} M$. The Lie derivative of a geometric object $Q$ of type $W$ with respect to a vector field $X$ on $M$ is defined as the geometric object ${\mathcal L} _ {X} Q$ of type $TW$( where $TW$ is the tangent bundle of $W$, regarded in a natural way as a $\mathop{\rm GL} ^ {k} ( n)$- space), given by the formula

$${\mathcal L} _ {X} Q = \left . \frac{d}{dt} Q \circ \phi _ {t} ^ {(} k) \right | _ {t=} 0 .$$

The value of the Lie derivative ${\mathcal L} _ {X} Q$ at a point $p _ {k} \in P ^ {k} M$ depends only on the $1$- jet of $Q$ at $p _ {k}$, and does so linearly, and on the value of $X ^ {(} k)$ at this point (or, equivalently, on the $k$- jet of $X$ at the corresponding point $x \in M$).

If the geometric object $Q$ is linear, that is, the corresponding $\mathop{\rm GL} ^ {k} ( n)$- space $W$ is a vector space with linear action of $\mathop{\rm GL} ^ {k} ( n)$, then the tangent manifold $TW$ can in a natural way be identified with the direct product $W \times W$, and so the Lie derivative

$${\mathcal L} _ {X} Q : P ^ {k} M \rightarrow T W = W \times W$$

can be regarded as a pair of geometric objects of type $W$. The first of these is $Q$ itself, and the second, which is usually identified with the Lie derivative of $Q$, is equal to the derivative $\partial _ {X ^ {(} k) } Q$ of $Q$ in the direction of the vector field $X ^ {(} k)$:

$${\mathcal L} _ {X} Q = ( Q , \partial _ {X ^ {(} k) } Q ) .$$

Thus, the Lie derivative of a linear geometric object can be regarded as a geometric object of the same type as $Q$.

Local coordinates $x ^ {i}$ in the manifold $M$ determine local coordinates $x ^ {i} , y _ {j} ^ {i}$ in the manifold $P ^ {1} M$ of coframes of order 1: for $\theta \in P ^ {1} M$ one has

$$\theta = \sum _ { j } y _ {j} ^ {i} d x ^ {j} .$$

In these coordinates the Lie derivative of any geometric object $Q = Q ( x ^ {i} , y _ {j} ^ {i)}$ of order 1 (for example, a tensor field) in the direction of the vector field

$$X = \sum _ { j } X ^ {j} \frac \partial {\partial x ^ {j} }$$

is given by the formula

$$( {\mathcal L} _ {X} Q ) ( x ^ {i} , y _ {j} ^ {i} ) = \ \sum _ { j } \frac \partial {\partial x ^ {i} } Q - \sum _ { i,j,l } y _ {l} ^ {i} X _ {j} ^ {l} \frac \partial {\partial y _ {j} ^ {i} } Q ,$$

where

$$X _ {j} ^ {l} = \frac \partial {\partial x ^ {j} } X ^ {l} .$$

A similar formula holds for the Lie derivative of a geometric object of arbitrary order.

The Lie derivative ${\mathcal L} _ {X}$ in the space of differential forms on a manifold $M$ can be expressed in terms of the operator of exterior differentiation $d$ and the operator of interior multiplication $i _ {X}$( defined as the contraction of a vector field with a differential form) by means of the following homotopy formula:

$${\mathcal L} _ {X} = d \circ i _ {X} + i _ {X} \circ d .$$

Conversely, the operator of exterior differentiation $d$, acting on a $p$- form $\omega$, can be expressed in terms of the Lie derivative by the formula

$$d \omega ( X _ {1} \dots X _ {p+} 1 ) =$$

$$= \ \sum _ { i= } 1 ^ { p+ } 1 (- 1) ^ {i+} 1 {\mathcal L} _ {X _ {i} } \omega ( X _ {1} \dots \widehat{X} _ {i} \dots X _ {p+} 1 ) +$$

$$+ \sum _ {i < j } (- 1) ^ {i+} j \omega ( {\mathcal L} _ {X _ {i} } X _ {j} , X _ {1} \dots \widehat{X} _ {i} \dots \widehat{X} _ {j} \dots X _ {p+} 1 ) ,$$

where $\widehat{ {}}$ means that the corresponding symbol must be omitted, and the $X _ {1} \dots X _ {p+} 1$ are vector fields.

In contrast to covariant differentiation, which requires the introduction of a connection, the operation of Lie differentiation is determined by the structure of the differentiable manifold, and the Lie derivative of a geometric object $Q$ in the direction of a vector field $X$ is a concomitant of the geometric objects $X$ and $Q$.

References

[1]	W. Slebodziński, "Sur les équations canonique de Hamilton" Bull. Cl. Sci. Acad. Roy. Belgique , 17 (1931) pp. 864–870
[2]	B.L. Laptev, "Lie differentiation" Progress in Math. , 6 (1970) pp. 229–269 Itogi. Nauk. Algebra Topol. Geom. 1965 (1967) pp. 429–465
[3]	K. Yano, "The theory of Lie derivatives and its applications" , North-Holland (1957)
[4]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
[5]	V.V. Vagner, "Theory of geometric objects and theory of finite and infinite continuous transformation groups" Dokl. Akad. Nauk SSSR , 46 (1945) pp. 347–349 (In Russian)
[6]	B.L. Laptev, "Lie derivative in a space of supporting elements" Trudy Sem. Vektor. Tenzor. Anal. , 10 (1956) pp. 227–248 (In Russian)
[7]	L.E. Evtushik, "The Lie derivative and differential field equations of a geometric object" Soviet Math. Dokl. , 1 (1960) pp. 687–690 Dokl. Akad. Nauk SSSR , 132 (1960) pp. 998–1001
[8]	R.S. Palais, "A definition of the exterior derivative in terms of Lie derivatives" Proc. Amer. Math. Soc. , 5 (1954) pp. 902–908