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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send feedback and suggestions to bombelli at olemiss.edu – modified 14 jan 2016