Vector
Fields on Manifolds| Vector
Fields on Manifolds |
In General > s.a. integration
on manifolds; lie
groups; vector calculus; vectors.
$ Def 1: A cross section of the tangent bundle TX.
$ Def 2: A derivation v: Ck(X) → Ck–1(X)
on the algebra of Ck functions on X taking
f 
Ck(X)
to v(f):= vf.
$ Divergence: Given a
volume element 
on M,
(div X)
= 
X ;
A convenient formula is
a va =
|g|–1/2 (|g|1/2 vm), m .
* Divergence (Gauss) theorem: 
M Da Xa dv =
bdry(M) Xa dSa
[@ Dray & Hellaby JMP(94)].
* Complex vector field: Can be visualized
as a field of ellipses [@ M Berry].
@ Generalizations: Chatterjee et al JPA(06)mp [and
forms]; Chatterjee & Lahiri a0705-in; > s.a. tangent
structures [second-order].
Special Types of Vector Fields > s.a. Projectable
Vector Field.
* Complete vector field: One that generates a global one-parameter
group
of transformations.
* Hypersurface-orthogonal: va is
hypersurface-orthogonal if v[a b vc]
= 0.
* Solenoidal: One whose divergence
vanishes, · v =
0.
$ Invariant: A vector field v TX
is invariant under the diffeomorphism f :
X → X if f '(x)vx = vf(x),
for all x X;
This
can also be written f 'v = v, or f*
v = v f*.
@ References: Hall CQG(06)
[covariantly constant, and curvature tensor].
Related Concepts > s.a. decomposition;
tangent structures [tangent map, push-forward].
* Flow: Given a C1 vector
field v on
a manifold X, the flow
of v on X is the mapping 
:
v → X given
by (x, t) 
(x, t),
where v:=
{(x, t)
| x X,
t I,
x = (t),
: I →
X} 
X × R.
* Twist: For a vector field va on a 4D manifold, the twist of va
is
a:= 
abcd vb c vd .
In a general n-dimensional manifold the twist, defined as above,
will be an (n–3)-form, but one may be able to define a 1-form
if there are preferred
vector fields
to contract
this with.
* Integral curve: Given v TX,
a curve :
I →
X, I R,
is an integral curve of v if d(t)/dt = v((t)).
* Moving frame: A set of n linearly independent differentiable
vector fields on X (of dimension n) which form a basis for
the module 
(U), U X; > s.a.
tetrad or vielbein.
@ Flow: Clark T&A(05)
[rotation class invariant].
Vector Bundle > s.a. tangent
structures.
$ Def: A fiber bundle whose fiber is a vector space.
* Examples: The tangent bundle
TM or cotangent bundle T*M of
any
manifold M.
$ Vertical vector field: A vector field in a fiber bundle is vertical
if
it is tangent to the fiber.
$ Vertical covector field:
Given a preferred horizontal subspace on a fiber bundle, a covector field is
vertical
if its contraction with any horizontal vector
vanishes.
* Remark: This can now be easily extended to a tensor field with any index
structure.
$ Linear vector field: A
projectable vector field v TX is
linear if every flow p is
a morphism of vector bundles over vt,
where p is
a flow of v, or p :
E → E, restricted to each fiber Ex is
a linear mapping p:
Ex → Esigma_t(x).
On a Lie Group
$ Invariant vector field:
The vector field v TG is
invariant under the left action Lg: G →
G if for all g, h G,
L'g(v(h))
=
v(Lg(h)) 
v(gh).
* Relationships: There
is a bijection between left invariant vector fields on G and
tangent vectors of G at the identity e.
* Remark: There is a
similar definition for right-invariant vector fields.
On Spaces with Other Structures > see poisson manifolds.
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send feedback and suggestions to bombelli at olemiss.edu – modified 21
aug
2009