Editing Tangent bundle (section)

==Canonical vector field on tangent bundle==
On every tangent bundle <math>TM</math>, considered as a manifold itself, one can define a '''canonical vector field''' <math>V:TM\rightarrow T^2M </math> as the [[diagonal map]] on the tangent space at each point. This is possible because the tangent space of a vector space ''W'' is naturally a product, <math>TW \cong W \times W,</math> since the vector space itself is flat, and thus has a natural diagonal map <math>W \to TW</math> given by <math>w \mapsto (w, w)</math> under this product structure. Applying this product structure to the tangent space at each point and globalizing yields the canonical vector field. Informally, although the manifold <math>M</math> is curved, each tangent space at a point <math>x</math>, <math>T_x M \approx \mathbb{R}^n</math>, is flat, so the tangent bundle manifold <math>TM</math> is locally a product of a curved <math>M</math> and a flat <math>\mathbb{R}^n.</math> Thus the tangent bundle of the tangent bundle is locally (using <math>\approx</math> for "choice of coordinates" and <math>\cong</math> for "natural identification"):

:<math>T(TM) \approx T(M \times \mathbb{R}^n) \cong TM \times T(\mathbb{R}^n)  \cong TM \times (  \mathbb{R}^n\times\mathbb{R}^n)</math>
and the map <math>TTM \to TM</math> is the projection onto the first coordinates:
:<math>(TM \to M) \times (\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n).</math>
Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.

If <math>(x,v)</math> are local coordinates for <math>TM</math>, the vector field has the expression

:<math> V = \sum_i \left. v^i \frac{\partial}{\partial v^i} \right|_{(x,v)}.</math>

More concisely, <math>(x, v) \mapsto (x, v, 0, v)</math> – the first pair of coordinates do not change because it is the section of a bundle and these are just the point in the base space: the last pair of coordinates are the section itself. This expression for the vector field depends only on <math>v</math>, not on <math>x</math>, as only the tangent directions can be naturally identified.

Alternatively, consider the scalar multiplication function:
:<math>\begin{cases}
\mathbb{R} \times TM \to TM \\
(t,v) \longmapsto tv
\end{cases}</math>

The derivative of this function with respect to the variable <math>\mathbb R</math> at time <math>t=1</math> is a function <math> V:TM\rightarrow T^2M </math>, which is an alternative description of the canonical vector field.

The existence of such a vector field on <math> TM </math> is analogous to the [[canonical one-form]] on the [[cotangent bundle]]. Sometimes <math> V </math> is also called the '''Liouville vector field''', or '''radial vector field'''. Using <math> V </math> one can characterize the tangent bundle. Essentially, <math> V </math> can be characterized using 4 axioms, and if a manifold has a vector field satisfying these axioms, then the manifold is a tangent bundle and the vector field is the canonical vector field on it. See for example, De León et al.