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 (even for the value of the Lie derivative at a point x we need more than the vector v^a at x), but not on a choice of connection or metric (it is a concomitant).
* Useful formula: For any p-form ω (p > 1),

£_v ω = v ·dω + d(v · ω) .

* 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 ω) = £_v(dω).

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

£_v f |_x:= lim_{t → 0} t⁻¹ [f(\(\sigma_t(x)\)) − 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 → 0} t⁻¹ [\(\sigma_t^{-1\prime}\)(w(σ_t(x))) −w(x)] ≡ [v, w]_x .

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

£_v ω|_x:= lim_{t → 0} t⁻¹ [σ_t*(ω(\(\sigma_t(x)\))) − ω(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 ∇_mv^a + M^a_m ∇_cv^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: It 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: The definition (given by Lichnerowicz for the case when v^a is a Killing vector field, where the explicit antisymmetrization in a and b is not necessary, and extended by Kosmann to the general case) is

£_v ψ:= v^a ∇_a ψ − \(1\over4\)(∇_[a v_b]) γ^aγ^b ψ ,

where the γs are the gamma matrices.

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 57; in Kolář et al 93; in Choquet-Bruhat & DeWitt-Morette 00.
@ For spinor fields: Lichnerowicz CRAS(63); Kosmann AMPA(71); Jhangiani FP(78), FP(78) [geometrical significance]; in Penrose & Rindler 86 [for infinitesimal conformal isometries]; Bilyalov TMP(92) [and Noether theorem for spinor fields]; Bourguignon & Gauduchon CMP(92) [and "metric Lie derivative"]; Delaney MS(93); Hurley & Vandyck JPA(94), JPA(94), JPA(95) [and covariant derivative]; Fatibene et al gq/96-proc; Ortín CQG(02)ht, comment Fatibene & Francaviglia a0904 [all spins]; Godina & Matteucci IJGMP(05)m.DG; Palese & Winterroth mp/06-proc; Sharipov a0801; Leão et al a1411 [geometrically motivated approach]; Helfer PRS(16)-a1602 [and fermion stress-energies]; > s.a. spin structure.
@ Generalized framework: Trautman in(72) [for maps between manifolds]; Hurley & Vandyck JMP(01); Godina & Matteucci JGP(03).
> Online resources: see Wikipedia page; MathWorld page [for spinors]; article available in A Trautman's webpage.

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