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: 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 ∈ TX 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 C¹ 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 ∈ 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 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 → E_σt(x).

On a Lie Group
$ Invariant vector field: The vector field v ∈ TG 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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