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nLab > Latest Changes: vertical vector field

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeMar 24th 2021
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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
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>T</mi><mi>P</mi><mspace width="0.27778em"/><mo>≃</mo><mspace width="0.27778em"/><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>π</mi> <mo>*</mo></msup><mi>T</mi><mi>B</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><msub><mo>⊕</mo> <mi>P</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>T</mi> <mi>π</mi></msub><mi>P</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> T P \;\simeq\; \big( \pi^\ast T B \big) \oplus_P \big( T_\pi P \big) </annotation></semantics></math>$
diff, v11, current
- 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
- 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
- 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
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeMar 27th 2021
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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
- 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_, Münster 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
- 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
- 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
- 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>P</mi><mo>≃</mo><msub><mi>T</mi> <mi>π</mi></msub><mi>P</mi><mo>⊕</mo><msup><mi>π</mi> <mo>*</mo></msup><mi>T</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">T P \simeq T_\pi P \oplus \pi^\ast T B</annotation></semantics></math>$ comes from the fact that the defining short exact sequence of vector bundles $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>T</mi> <mi>π</mi></msub><mi>P</mi><mo>⟶</mo><mi>T</mi><mi>P</mi><mo>⟶</mo><msup><mi>π</mi> <mo>*</mo></msup><mi>T</mi><mi>B</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to T_\pi P \longrightarrow T P \longrightarrow \pi^\ast T B \to 0</annotation></semantics></math>$ splits.

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

diff, v17, current

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nLab > Latest Changes: vertical vector field