Lie Derivatives  

In General > s.a. Derivatives.
* Idea: A notion of directional derivative on an arbitrary differentiable manifold that depends on a vector field va (even for the value of the Lie derivative at a point x we need more than the vector va 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) = −(∂vj/∂xi) (∂/∂xj) ,   £v (dxi) = (∂vi/∂xj) dxj .

* 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:= limt → 0 t−1 [f(\(\sigma_t(x)\)) − f(x)] = v(f)|xvaa f |x .

* Vector fields: Defined using the push-forward under the diffeomorphisms generated by va,

£v w|x:= limt → 0 t−1 [\(\sigma_t^{-1\prime}\)(w(σt(x))) −w(x)] ≡ [v, w]x .

* One-forms: Defined using the pull-back under the diffeomorphisms generated by va,

£v ω|x:= limt → 0 t−1 [σt*(ω(\(\sigma_t(x)\))) − ω(x)] .

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

£v Mac:= vmm MacMmcmva + Mamcvm .

* Scalar / tensor densities of weight 1:

£v f = vaa f + fa va = ∂a (va f) ;   £v M = |g|1/2 £v M + (div v) M .

* With torsion: It includes additional terms; For example,

£v Xa = vmm XaXmm vavm Xn Tmna .

* Spinor fields: The definition (given by Lichnerowicz for the case when va 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:= vaa ψ − \(1\over4\)(∇[a vb]) γaγb 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].

@ 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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