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__:
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}
∇_{m}*v*^{a}
+ *M*^{a}_{m}
∇_{c}*v*^{m} .

* __Scalar / tensor densities of weight 1__:

£_{v} * f*
=

* __With torsion__: It 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(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.

main page
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send feedback and suggestions to bombelli at olemiss.edu – modified 17 aug 2019