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.
* Complex vector field:
It can be visualized as a field of ellipses [@ M Berry].
@ Generalizations: Chatterjee et al JPA(06)mp [and forms];
Chatterjee & Lahiri a0705-conf;
> s.a. tangent structures [second-order].
> In physics: see low-spin field theories.
> Online resources:
see MathWorld page
[2015.11, the (y, x) and (–y, x) fields are incorrectly drawn];
Wikipedia page.
Special Types of Vector Fields > s.a. decomposition
of tensor fields; 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; > s.a. Newman-Penrose formalism.
* 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*
\(\circ\) v = v \(\circ\) f*.
@ References: Hall CQG(06) [covariantly constant, and curvature tensor].
Related Concepts
> s.a. 2D manifolds [Lie elgebras of vector fields]; Flux;
tangent structures [tangent map, push-forward]; vector calculus.
* Flow: Given a C1
vector field v on a manifold X, the flow of v on X is the
mapping σ: Σv → X
given by (x, t) \(\mapsto\) σ(x, t),
where Σv:= {(x, t)
| x ∈ X, t ∈ I, x = σ(t),
σ: I → X} ⊂ X × \(\mathbb R\).
* Integral curve: Given v ∈ TX,
a curve σ: I → X, I ⊂ \(\mathbb R\), is an integral curve of
v if dσ(t)/dt = v(σ(t));
The orbit is the image σ(I) ⊂ X.
* Weinstein conjecture:
Every Reeb vector field on a closed oriented three-manifold has a closed orbit;
Proved by Taubes using Seiberg-Witten theory; > s.a.
Wikipedia page.
* Moving frame: A set of n linearly independent
differentiable vector fields on X (of dimension n) which form a basis for the module
χ(U), U ⊂ X; > s.a. tetrad or vielbein.
@ References: Clark T&A(05) [topological invariant for flows];
Hutchings BAMS(10) [3D Weinstein conjecture, Taubes proof].
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 σp is a morphism
of vector bundles over vt,
where σp is a flow of
v, or σp : E
→ E, restricted to each fiber Ex
is a linear mapping σp:
Ex →
Eσt(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.
main page
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 22 jan 2016