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