Lie differentiation

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

In the special case when is a vector-valued function on , its Lie derivative coincides with the derivative of the function in the direction of the vector field and is given by the formula

where is the one-parameter local transformation group on generated by , or, in the local coordinates , by the formula

where

In the general case the definition of Lie differentiation consists in the following. Let be a -space, that is, a manifold with a fixed action of the general differential group of order (the group of -jets at the origin of diffeomorphisms , ). Let be a geometric object of order and type on an -dimensional manifold , regarded as a -equivariant mapping of the principal -bundle of coframes of order on into . The one-parameter local transformation group on generated by a vector field on induces a one-parameter local transformation group on the manifold of coframes . Its velocity field

is called the complete lift of to . The Lie derivative of a geometric object of type with respect to a vector field on is defined as the geometric object of type (where is the tangent bundle of , regarded in a natural way as a -space), given by the formula

The value of the Lie derivative at a point depends only on the -jet of at , and does so linearly, and on the value of at this point (or, equivalently, on the -jet of at the corresponding point ).

If the geometric object is linear, that is, the corresponding -space is a vector space with linear action of , then the tangent manifold can in a natural way be identified with the direct product , and so the Lie derivative

can be regarded as a pair of geometric objects of type . The first of these is itself, and the second, which is usually identified with the Lie derivative of , is equal to the derivative of in the direction of the vector field :

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

Local coordinates in the manifold determine local coordinates in the manifold of coframes of order 1: for one has

In these coordinates the Lie derivative of any geometric object of order 1 (for example, a tensor field) in the direction of the vector field

is given by the formula

where

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

The Lie derivative in the space of differential forms on a manifold can be expressed in terms of the operator of exterior differentiation and the operator of interior multiplication (defined as the contraction of a vector field with a differential form) by means of the following homotopy formula:

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

where means that the corresponding symbol must be omitted, and the 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 in the direction of a vector field is a concomitant of the geometric objects and .

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