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* · d*ω* + d(*v* · *ω*) .

* __Lie derivatives of the coordinate basis elements__:

£_{v}(∂/∂*x*^{i})
= –(∂*v*^{j}/∂*x*^{i})
(∂/∂*x*^{j})
, £_{v}
(d*x*^{i}) = (∂*v*^{i}/∂*x*^{j})
d*x*^{j} .

* __Other properties__:
Acting on forms, it commutes with taking the exterior derivative, d(£_{v} *ω*)
= £_{v}(d*ω*).

**For Various Types of Fields** > s.a. spin
coefficients; spin structures.

* __Scalar functions__: Coincides with other notions of derivative,

£_{v}* f* |_{x}:=
lim_{t → 0}* t*^{–1}
[*f*(*σ*_{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} [*σ*_{t}^{–1}*'*(*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*(*σ*_{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} ∇_{m}*v*^{a}
+ *M*^{a}_{m} ∇_{c}*v*^{m}
.

* __Scalar / tensor densities of weight 1__:

£_{v} * f* =

* __With torsion__: Includes additional terms; For example,

£_{v}* X*^{a}
= *v*^{m} ∇_{m}* X*^{a} –
*X*^{m} ∇_{m}* v*^{a} –* v*^{m}* X*^{n}* 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}*
∇

where the *γ*s 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-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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send feedback and suggestions to bombelli at olemiss.edu – modified 3
mar
2016