Tangent Structures to a Manifold  

Tangent Vector at a Point > s.a. vector; vector calculus; vector field.
$ Def: There are various possible definitions, including:
(1) A derivation on the algebra of germs of differentiable functions at xX;
(2) An equivalence class of triples (x, φ, V), with (x, φ', V') ~ (x, φ, V) if V' = D(φ' \(\circ\) φ–1)|x V (i.e., V transforms like a vector);
(3) An equivalence class of curves, tangent to each other at x.
> Online resources: see MathWorld page; Wikipedia page on tangent vector and tangent space.

Tangent Bundle
$ Def: The set TM of all tangent vectors at all points of an n-dimensional manifold M, with a differentiable fiber bundle structure.
* Fibers: The tangent spaces TpM at each pM; Structure group: GL(n, \(\mathbb R\)).
* Coordinates: Given coordinates {xi} on M, natural coordinates on TM are {xi, ∂/∂xi}.
* Relationships: It is an associated bundle to the frame bundle FM of a manifold M, with structure group GL(n, \(\mathbb R\)).
@ References: Yano & Ishihara 73; Morandi et al PRP(90); Hindeleh 09 [of Lie groups].
> Online resources: see Wikipedia page.

Related Concepts > s.a. Jet and Jet Bundle; tensor; tensor field.
* Distribution: A distribution S of dimension r on M is an assignment, to each pM, of an r-dimensional subspace Sp ⊂ TpM; Involutive distribution: A distribution S such that for all X, YS, [X, Y] ∈ S.
* Push-forward map: Given a map f : MN between differentiable manifolds, the pushforward f ' or f* is a map between vector fields.
* Tangent map: Given a map f : MN between differentiable manifolds, the tangent map Tf is a map between vectors (elements of TM and TN).

Cotangent Structures > s.a. differential forms.
$ Cotangent vector: A cotangent vector at a point pM is a dual vector, i.e., a map ω: TpM → \(\mathbb R\) from vectors to the reals.
$ Cotangent bundle: The set T*M of all cotangent vectors at all points of an n-dimensional manifold M, with a differentiable fiber bundle structure.

Generalizations > s.a. Topological Tangent Bundle.
@ Second-order tangent structures: Dodson & Galanis JGP(04) [infinite-dimensional manifolds].
@ Related topics: in Boroojerdian IJTP(13)-a1211 [\(\mathbb Z\)2-graded tangent bundle].


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