Lie Derivatives

In General > s.a. Derivatives.
* Idea: A notion of directional derivative on an arbitrary differentiable manifold that depends on a vector field v^a, but not on a choice of connection (it is a concomitant).
* Useful formula: For any p-form (p > 1),

_v = v · d + d(v · ) .

* Lie derivatives of the coordinate basis elements:

_v(/xⁱ) = –(v^j/xⁱ) (/x^j) ,   _v (dxⁱ) = (vⁱ/x^j) dx^j .

* Other properties: Acting on forms, it commutes with taking the exterior derivative, d(_v ) = _v(d).

For Various Types of Fields > s.a. spin coefficients; spin structures.
* Scalar functions: Coincides with other notions of derivative,

_v f |_x:= lim_{t to 0} t^–1 [f(_t(x)) – f(x)] = v(f)|_x v^a _a f |_x .

* Vector fields: Defined using the push-forward under the diffeomorphisms generated by v^a,

_v w|_x:= lim_{t to 0} t^–1 [_t^–1'(w(_t(x))) – w(x)] = [v,w]_x .

* One-forms: Defined using the pull-back under the diffeomorphisms generated by v^a,

_v |_x:= lim_{t to 0} t^–1 [_t*((_t(x))) – (x)] .

* Arbitrary tensor fields: Defined implicitly by the product rule _v (M N) = _v M N + M _v N; e.g.,

_v M^a_c:= v^m _m M^a_c – M^m_{c m}v^a + M^a_{m c}v^m .

* Scalar/tensor densities of weight 1:

_v f = v^a _a f + f _a v^a = _a (v^a f) ;   _v M = |g|^1/2 _v M + (div v) M .

* With torsion: Includes additional terms; For example,

_v X^a = v^m _m X^a – X^m _m v^a – v^m Xⁿ T_mn^a .

* Spinor fields: Defined only if v^a is a Killing vector field, otherwise one would have to define some Lie derivative of a pair spinor-metric somehow; It is given by _v := v^a _a + (_a k_b) ^ab .

Related Concepts
* Lie bracket: The Lie derivative induces a Lie bracket structure on vector fields,

[v, w] = _v w .

@ References: Crainic & Fernandes AM(03) [integrability].

References
@ General: Schouten 54 [red threads embedded in gelatin]; Yano 55.
@ For spinor fields: Kosmann AMPA(71); Hurley & Vandyck JPA(94), JPA(94), JPA(95) [and covariant derivative]; Fatibene et al gq/96-in; Ortín CQG(02)ht [all spins]; Godina & Matteucci IJGMP(05)m.DG.
@ Generalized framework: Hurley & Vandyck JMP(01).
> Online resources: see Wikipedia page.

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