Lie derivative
This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.
Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function.
Generalisations exist for spinor fields, fibre bundles with a connection and vector-valued differential forms.
In differential geometry, there are three main coordinate independent notions of differentiation of tensor fields: 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 vector is well-defined even if it is not specified how to extend that tangent vector to a vector field.
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.
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).
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.
is the unique solution of the system of first-order autonomous (i.e. time-independent) differential equations, with
Consider T to be a differentiable multilinear map of smooth sections α1, α2, ..., αp of the cotangent bundle T∗M and of sections X1, X2, ..., Xq of the tangent bundle TM, written T(α1, α2, ..., X1, X2, ...) into R. Define the Lie derivative of T along Y by the formula The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the Leibniz rule for differentiation.
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.
The Cartan formula can be used as a definition of the Lie derivative of a differential form.
Cartan's formula shows in particular that The Lie derivative also satisfies the relation In local coordinate notation, for a type (r, s) tensor field
is the set of vector fields on M:[4] which may also be written in the equivalent notation where the tensor product symbol
is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold.
Thus, for example, considered as a derivation on a vector field, one finds the above to be just the Jacobi identity.
The Lie derivative also has important properties when acting on differential forms.
Then Various generalizations of the Lie derivative play an important role in differential geometry.
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.
[5] Later, it was provided a geometric framework which justifies her ad hoc prescription within the general framework of Lie derivatives on fiber bundles[6] in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.
admitting a spin structure, the Lie derivative of a spinor field
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,[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 the Kosmann lift.
Another generalization, due to Albert Nijenhuis, allows one to define the Lie derivative of a differential form along any section of the bundle Ωk(M, TM) of differential forms with values in the tangent bundle.
The Nijenhuis–Lie derivative is then the anticommutator of the interior product and the exterior derivative: 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 fiber bundles) were studied by A. Nijenhuis, Y. Tashiro and K. Yano.
For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians.
induced by an infinitesimal transformation of coordinates generated by a vector field
