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 *x* ∈ *X*;

(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 T*M* of
all tangent vectors at all points of an *n*-dimensional
manifold *M*, with a differentiable fiber bundle structure.

* __Fibers__: The tangent
spaces T_{p}*M* at each *p* ∈ *M*; __Structure group__: GL(*n*, \(\mathbb R\)).

* __Coordinates__: Given coordinates {*x*^{i}}
on *M*, natural coordinates
on T*M* are {*x*^{i}, ∂/∂*x*^{i}}.

* __Relationships__: It is
an associated bundle to the frame bundle F*M* 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 *p* ∈ *M*,
of an *r*-dimensional subspace
*S*_{p} ⊂ T_{p}*M*;
__Involutive distribution__: A distribution *S* such that for all *X*, *Y* ∈ *S*,
[*X*,* Y*] ∈ *S*.

* __Push-forward map__: Given
a map *f* : *M* → *N* between differentiable manifolds, the pushforward
*f '* or *f*_{*} is a map between vector fields.

* __Tangent map__: Given a
map *f* : *M* → *N* between differentiable manifolds,
the tangent map T*f* is a map between vectors (elements of T*M* and T*N*).

**Cotangent Structures** > s.a. differential forms.

$ __Cotangent vector__: A cotangent vector at a point *p* ∈ *M* is a dual vector, i.e., a map *ω*: T_{p}*M* → \(\mathbb R\) 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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jan
2016