nLab vertical vector field

Redirected from "vertical tangent bundles".

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)