# nLab vertical vector field

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Definition

### Vertical vector fields

Let $\pi : P \to X$ be a bundle in the category of SmoothManifolds. A vector field $v \in \Gamma(T P)$ is vertical with respect to this bundle if it is in the kernel of the derivative $d \pi \colon T P \to T X$.

### Vertical tangent bundle

The collection of vertical vectors forms the vertical tangent bundle inside the full tangent bundle, typically denoted $T_\pi P$.

For $\pi \colon P \to B$ a smooth function between smooth manifolds, its vertical tangent bundle is the fiber-wise kernel of the differential $d \pi$, as shown in the following diagram (e.g Tu 17 (27.4), Berglund 20, p. 16):

### Horizontal differential forms

A differential form on $P$ is a horizontal differential form with respect to $P \to X$ it it vanishes on vertical vector fields.

## Properties

###### Proposition

If $P \overset{\pi}{\longrightarrow} X$ is a surjective submersion (for instance a smooth fiber bundle) then the full tangent bundle of its total space $P$ is isomorphic to the direct sum of the vertical tangent bundle (above) with the pullback of the tangent bundle of the base space:

$T P \;\simeq\; \big( \pi^\ast T B \big) \oplus_P \big( T_\pi P \big)$

###### Proof

The assumption that $\pi$ is a surjective submersion implies that $d \pi \colon T P \longrightarrow \pi^\ast T B$ is a surjection and hence that

$0 \to T_\pi P \longrightarrow T P \overset{d \pi}{\longrightarrow} \pi^\ast T B \to 0$

is a short exact sequence of smooth real vector bundles.

Now all short exact sequences of real vector bundles over paracompact topological spaces (such as smooth manifolds) split (by a choice of fiberwise metric, see at short exact sequence of vector bundles), which is the statement to be shown.

## Examples

###### Proposition

(vertical tangent bundle of a real vector bundle) Let $\pi \colon \mathcal{V} \longrightarrow M$ be a real vector bundle. Then its vertical tangent bundle is isomorphic to the its pullback along itself:

$T_\pi \mathcal{V} \;\simeq_{{}_{\mathcal{V}}}\; \pi^\ast \mathcal{V} \,.$

###### Proposition

(stable tangent bundle of unit sphere bundle)
The once-stabilized tangent bundle of a unit sphere bundle $S(\mathcal{V})$ in a real vector bundle $\mathcal{V} \overset{p}{\longrightarrow} M$ (Example ) over a smooth manifold $M$ is isomorphic to the pullback of the direct sum of the stable tangent bundle of the base manifold with that vector bundle:

$T S(\mathcal{V}) \times \mathbb{R} \; \simeq_{{}_M} \; S(p)^\ast \big( T M \oplus_M \mathcal{V} \big) \,.$

This is stated without proof as Crowley-Escher 03, Fact. 3.1, apparently reading between the lines in Milnor 56, p. 403. The key sub-statement that

$T_{S(p)} S(\mathcal{V}) \times \mathbb{R} \;\simeq_{{}_M}\; \mathcal{V}$

is made explicit in Gollinger 16, Prop. 1.1.9

###### Proof

Consider first the actual tangent bundle but to the open ball/disk-fiber bundle $D(\mathcal{V})$ that fills the given sphere-fiber bundle: By the standard splitting (this Prop.) this is the direct sum

$T \big( D(\mathcal{V}) \big) \;\simeq\; \big( D(p)^\ast T M \big) \oplus_M T_p D(\mathcal{V}) \,,$

where $T_p D(\mathcal{V})$ is the vertical tangent bundle of the disk bundle. But, by definition of disk bundles, that is the restriction of the vertical tangent bundle of the vector bundle $\mathcal{V}$ itself, and that is just the pullback of that vector bundle along itself (by this Example):

\begin{aligned} \cdots & \simeq\; \big( D(p)^\ast T M \big) \oplus_M \big( D(p)^\ast \mathcal{V} \big) \\ & \simeq\; D(p)^\ast \big( T M \oplus_M \mathcal{V} \big) \,. \end{aligned}

To conclude, it just remains to observe that the normal bundle of the n-sphere-boundary inside the $(n+1)$-ball is manifestly trivial, so that the restriction of the tangent bundle of $D(\mathcal{V})$ to $S(\mathcal{V})$ is the stable tangent bundle of $S(\mathcal{V})$.

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