synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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 $d \pi \colon T P \to T X$.
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):
A differential form on $P$ is a horizontal differential form with respect to $P \to X$ it it vanishes on vertical vector fields.
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:
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
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.
(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:
(e.g. tomDieck 00 (6.9), tomDieck 08 (15.6.7), Gollinger 16, inside the proof of Prop. 1.1.9)
(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:
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
is made explicit in Gollinger 16, Prop. 1.1.9
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
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):
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 $S(\mathcal{V})$.
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)
See also:
Tammo tom Dieck, around Satz 6.9 in: Topologie, De Gruyter (2000)(doi:10.1515/9783110802542)
Tammo tom Dieck, around (15.6.7) Algebraic topology, European Mathematical Society, Zürich (2008) (doi:10.4171/048, pdf)
William Gollinger, Section 1.1.4 in: Madsen-Tillmann-Weiss Spectra and a Signature Problem for Manifolds, Münster 2016 (pdf, pdf)
Alexander Berglund, Characteristic classes for families of bundles (arXiv:2012.12170)
Last revised on June 4, 2022 at 16:49:35. See the history of this page for a list of all contributions to it.