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 T*X*.

$ __Def 2__: A derivation *v*: C^{k}(*X*) → C^{k–1}(*X*)
on the algebra of C^{k} functions on *X* taking
*f* ∈ C^{k}(*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__: *v*^{a} is
hypersurface-orthogonal if *v*_{[a} ∇_{b} *v*_{c]}
= 0; > s.a. Newman-Penrose formalism.

* __Solenoidal__: One whose divergence
vanishes, ∇ · **v** = 0.

$ __Invariant__: A vector field *v* ∈ T*X*
is invariant under the diffeomorphism* f* :
*X* → *X* if *f* '(*x*)*v*_{x}
= *v*_{f(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 C^{1} vector
field *v* on a manifold *X*, the flow
of *v* on *X* is the mapping *σ*:
Σ_{v }→ *X* given
by (*x*, *t*) \(\mapsto\) *σ*(*x*, *t*),
where Σ_{v}:= {(*x*, *t*)
| *x* ∈ *X*, *t* ∈ I,
*x* = *σ*(*t*), *σ*: I →
*X*} ⊂ *X* × \(\mathbb R\).

* __Integral curve__: Given *v* ∈ T*X*,
a curve *σ*: I → *X*, I ⊂ \(\mathbb R\),
is an integral curve of *v* if d*σ*(*t*)/d*t* = *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
T*M* 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* ∈ T*X* is
linear if every flow *σ*_{p} is
a morphism of vector bundles over *v*_{t},
where *σ*_{p} is
a flow of *v*, or *σ*_{p} :
*E* → *E*, restricted to each fiber *E*_{x} is
a linear mapping *σ*_{p}:
*E _{x}* →

**On a Lie Group**

$ __Invariant vector field__: The vector field *v* ∈ T*G*
is invariant under the left action L_{g}: *G* →
*G* if for all *g*, *h* ∈ *G*,
L*'*_{g}(*v*(*h*))
= *v*(L_{g}(*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 22
jan
2016