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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 24th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded brief mentioning of the vertical tangent bundle and statement of the splitting formula $$ T P \;\simeq\; \big( \pi^\ast T B \big) \oplus_P \big( T_\pi P \big) $$ <a href="https://ncatlab.org/nlab/revision/diff/vertical+vector+field/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/vertical+vector+field/11">v11</a>, <a href="https://ncatlab.org/nlab/show/vertical+vector+field">current</a>

   added brief mentioning of the vertical tangent bundle and statement of the splitting formula

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

   diff, v11, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 24th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded more of an actual definition of the vertical tangent bundle ([here](https://ncatlab.org/nlab/show/vertical+vector+field#VerticalTangentBundle), need to give a more canonical reference...) <a href="https://ncatlab.org/nlab/revision/diff/vertical+vector+field/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/vertical+vector+field/11">v11</a>, <a href="https://ncatlab.org/nlab/show/vertical+vector+field">current</a>

   added more of an actual definition of the vertical tangent bundle (here, need to give a more canonical reference…)

   diff, v11, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 24th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement of the example of the vertical tangent bundle of a vector bundle ([here](https://ncatlab.org/nlab/show/vertical+vector+field#VerticalTangentBundleOfRealVectorBundle)) <a href="https://ncatlab.org/nlab/revision/diff/vertical+vector+field/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/vertical+vector+field/12">v12</a>, <a href="https://ncatlab.org/nlab/show/vertical+vector+field">current</a>

   added statement of the example of the vertical tangent bundle of a vector bundle (here)

   diff, v12, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 24th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Loring Tu]], Section 27.5 in: _Differential Geometry -- Connections, Curvature, and Characteristic Classes_, Springer 2017 ([ISBN:978-3-319-55082-4](https://www.springer.com/gp/book/9783319550824)) <a href="https://ncatlab.org/nlab/revision/diff/vertical+vector+field/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/vertical+vector+field/13">v13</a>, <a href="https://ncatlab.org/nlab/show/vertical+vector+field">current</a>

   added pointer to:

   • Loring Tu, Section 27.5 in: Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer 2017 (ISBN:978-3-319-55082-4)

   diff, v13, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 27th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[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](https://www.elsevier.com/books/connections-curvature-and-cohomology-v1/greub/978-0-12-302701-6)) <a href="https://ncatlab.org/nlab/revision/diff/vertical+vector+field/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/vertical+vector+field/15">v15</a>, <a href="https://ncatlab.org/nlab/show/vertical+vector+field">current</a>

   added pointer to:

   • 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)

   diff, v15, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 27th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * William Gollinger, Section 1.1.4 in: _Madsen-Tillmann-Weiss Spectra and a Signature Problem for Manifolds_, M&uuml;nster 2016 ([pdf](https://repositorium.uni-muenster.de/document/miami/7369e8b5-6ae4-4e42-b6f3-4602ec24427a/diss_gollinger.pdf)) as one place where the characterization of the vertical tangent bundle of a sphere-fiber bundle is made explicit <a href="https://ncatlab.org/nlab/revision/diff/vertical+vector+field/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/vertical+vector+field/15">v15</a>, <a href="https://ncatlab.org/nlab/show/vertical+vector+field">current</a>

   added pointer to

   • William Gollinger, Section 1.1.4 in: Madsen-Tillmann-Weiss Spectra and a Signature Problem for Manifolds, Münster 2016 (pdf)

   as one place where the characterization of the vertical tangent bundle of a sphere-fiber bundle is made explicit

   diff, v15, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Tammo tom Dieck]], around (15.6.7) of: _Algebraic topology_, European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/vertical+vector+field/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/vertical+vector+field/17">v17</a>, <a href="https://ncatlab.org/nlab/show/vertical+vector+field">current</a>

   added pointer to

   • Tammo tom Dieck, around (15.6.7) of: Algebraic topology, European Mathematical Society, Zürich (2008) (doi:10.4171/048, pdf)

   diff, v17, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Tammo tom Dieck]], around Satz 6.9 in: _Topologie_, De Gruyter (2000)([doi:10.1515/9783110802542](https://doi.org/10.1515/9783110802542)) <a href="https://ncatlab.org/nlab/revision/diff/vertical+vector+field/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/vertical+vector+field/17">v17</a>, <a href="https://ncatlab.org/nlab/show/vertical+vector+field">current</a>

   added pointer to:

   • Tammo tom Dieck, around Satz 6.9 in: Topologie, De Gruyter (2000)(doi:10.1515/9783110802542)

   diff, v17, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have made more explicit how the decomposition $T P \simeq T_\pi P \oplus \pi^\ast T B$ comes from the fact that the defining short exact sequence of vector bundles $0 \to T_\pi P \longrightarrow T P \longrightarrow \pi^\ast T B \to 0$ splits. Maybe we should have an entry _[[short exact sequence of vector bundles]]_. <a href="https://ncatlab.org/nlab/revision/diff/vertical+vector+field/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/vertical+vector+field/17">v17</a>, <a href="https://ncatlab.org/nlab/show/vertical+vector+field">current</a>

   I have made more explicit how the decomposition $T P \simeq T_\pi P \oplus \pi^\ast T B$ comes from the fact that the defining short exact sequence of vector bundles $0 \to T_\pi P \longrightarrow T P \longrightarrow \pi^\ast T B \to 0$ splits.

   Maybe we should have an entry short exact sequence of vector bundles.

   diff, v17, current