nLab vertical vector field

Contents

Context

Differential geometry

Definition

Vertical vector fields
Vertical tangent bundle
Horizontal differential forms

Properties
Examples
Related concepts
References

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} \,.

(e.g. tomDieck 00 (6.9), tomDieck 08 (15.6.7), Gollinger 16, inside the proof of Prop. 1.1.9)

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 $S(\mathcal{V})$ .

Related concepts

evolutionary vector field

References

Textbook accounts:

Werner Greub, Stephen Halperin, Ray Vanstone, Section VII.1 in Volume 1 De Rham Cohomology of Manifolds and Vector Bundles, in: Connections, Curvature, and Cohomology Academic Press (1973) (ISBN:978-0-12-302701-6)
Loring Tu, Section 27.5 in: Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer 2017 (ISBN:978-3-319-55082-4, pdf)