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nLab > Latest Changes: Ehresmann connection

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeSep 23rd 2010
- PermaLink
Author: Urs
Format: MarkdownItexpolished and expanded [[Ehresmann connection]]

polished and expanded Ehresmann connection
- CommentRowNumber2.
- CommentAuthorzskoda
- CommentTimeNov 30th 2011
- (edited Nov 30th 2011)
- PermaLink
Author: zskoda
Format: MarkdownItexI have improved (hopefully) the section on the definition via horizontal subspaces at [[Ehresmann connection]]. On the other hand, I think (and [wikipedia](http://en.wikipedia.org/wiki/Ehresmann_connection) agrees) that the statement about the terminology is wrong at two places. One is the statement in the entry that the Ehresmann connection must be on a principal bundle (but must be on a fiber bundle) to be called such and another is suspicious phrase "Cartan-Ehresmann connection", in my opinion [[Cartan connection]] is by the definition in a smaller generality then Ehresmann. Finally the Ehresmann connections on a principal and its associated bundles are in 1-1 correspondence: If $T^H P\subset T P$ is the smooth horizontal distrubution of subspaces defining the principal connection on a principal $G$-bundle $P$ over $X$, where $G$ is a Lie group and $F$ a smooth left $G$-space, then consider the total space $P\times_G F$ of the associated bundle with typical fiber $F$. Then, for a fixed $f\in F$ one defines a map $\rho_f : P\to P\times_G F$ assigning the class $[p,f]$ to $p\in P$. If $(T_p \rho_f)(T^H_p P) =: T_{[p,f]}^H P\times_G F$ defines the horizontal subspace $T_{[p,f]}^H P\times_G F\subset T_{[p,f]} P\times_G F$, the collection of such subspaces does not depend on the choice of $(p,f)$ in the class $[p,f]$, and the correspondence $p\mapsto T_{[p,f]}^H P\times_G F$ is a connection on the associated bundle $P\times_G F\to X$. I added now this reasoning to the entry as well.

I have improved (hopefully) the section on the definition via horizontal subspaces at Ehresmann connection. On the other hand, I think (and wikipedia agrees) that the statement about the terminology is wrong at two places. One is the statement in the entry that the Ehresmann connection must be on a principal bundle (but must be on a fiber bundle) to be called such and another is suspicious phrase “Cartan-Ehresmann connection”, in my opinion Cartan connection is by the definition in a smaller generality then Ehresmann.

Finally the Ehresmann connections on a principal and its associated bundles are in 1-1 correspondence: If $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>T</mi> <mi>H</mi></msup><mi>P</mi><mo>⊂</mo><mi>T</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">T^H P\subset T P</annotation></semantics></math>$ is the smooth horizontal distrubution of subspaces defining the principal connection on a principal $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ -bundle $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>$ over $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ , where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ is a Lie group and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>$ a smooth left $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ -space, then consider the total space $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">P\times_G F</annotation></semantics></math>$ of the associated bundle with typical fiber $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>$ . Then, for a fixed $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">f\in F</annotation></semantics></math>$ one defines a map $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi> <mi>f</mi></msub><mo>:</mo><mi>P</mi><mo>→</mo><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">\rho_f : P\to P\times_G F</annotation></semantics></math>$ assigning the class $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>p</mi><mo>,</mo><mi>f</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[p,f]</annotation></semantics></math>$ to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">p\in P</annotation></semantics></math>$ . If $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>T</mi> <mi>p</mi></msub><msub><mi>ρ</mi> <mi>f</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>T</mi> <mi>p</mi> <mi>H</mi></msubsup><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><mo>:</mo><msubsup><mi>T</mi> <mrow><mo stretchy="false">[</mo><mi>p</mi><mo>,</mo><mi>f</mi><mo stretchy="false">]</mo></mrow> <mi>H</mi></msubsup><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">(T_p \rho_f)(T^H_p P) =: T_{[p,f]}^H P\times_G F</annotation></semantics></math>$ defines the horizontal subspace $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>T</mi> <mrow><mo stretchy="false">[</mo><mi>p</mi><mo>,</mo><mi>f</mi><mo stretchy="false">]</mo></mrow> <mi>H</mi></msubsup><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>F</mi><mo>⊂</mo><msub><mi>T</mi> <mrow><mo stretchy="false">[</mo><mi>p</mi><mo>,</mo><mi>f</mi><mo stretchy="false">]</mo></mrow></msub><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">T_{[p,f]}^H P\times_G F\subset T_{[p,f]} P\times_G F</annotation></semantics></math>$ , the collection of such subspaces does not depend on the choice of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,f)</annotation></semantics></math>$ in the class $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">[</mo><mi>p</mi><mo>,</mo><mi>f</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[p,f]</annotation></semantics></math>$ , and the correspondence $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>↦</mo><msubsup><mi>T</mi> <mrow><mo stretchy="false">[</mo><mi>p</mi><mo>,</mo><mi>f</mi><mo stretchy="false">]</mo></mrow> <mi>H</mi></msubsup><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">p\mapsto T_{[p,f]}^H P\times_G F</annotation></semantics></math>$ is a connection on the associated bundle $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>F</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P\times_G F\to X</annotation></semantics></math>$ . I added now this reasoning to the entry as well.
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeJun 28th 2012
- (edited Jun 28th 2012)
- PermaLink
Author: Urs
Format: MarkdownItexI have added to _[[Ehresmann connection]]_ a pointer to the formalization of flat Ehresmann connections in [[cohesive homotopy type theory]]. I have just posted a little more chat about this _[here](http://golem.ph.utexas.edu/category/2012/06/flat_ehresmann_connections_in.html)_ to the $n$Café.

I have added to Ehresmann connection a pointer to the formalization of flat Ehresmann connections in cohesive homotopy type theory.

I have just posted a little more chat about this here to the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ Café.
- CommentRowNumber4.
- CommentAuthorDmitri Pavlov
- CommentTimeFeb 14th 2021
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexAdded a link to the PDF file of the original paper. <a href="https://ncatlab.org/nlab/revision/diff/Ehresmann+connection/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/Ehresmann+connection/19">v19</a>, <a href="https://ncatlab.org/nlab/show/Ehresmann+connection">current</a>

Added a link to the PDF file of the original paper.

diff, v19, current
- CommentRowNumber5.
- CommentAuthorzskoda
- CommentTimeFeb 16th 2021
- PermaLink
Author: zskoda
Format: MarkdownItexUrs, 3: link to $n$Café seem not to work.

Urs, 3: link to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ Café seem not to work.
- CommentRowNumber6.
- CommentAuthorTim_Porter
- CommentTimeFeb 16th 2021
- (edited Feb 16th 2021)
- PermaLink
Author: Tim_Porter
Format: MarkdownItexI wrote: >Even if you go to the café and find that [day's entry](https://golem.ph.utexas.edu/category/2012/06/index.shtml), the follow up gives the same error. The problem is thus in the Café. but when I checked back the problem did not seem to exist any more. Strange.

I wrote:

Even if you go to the café and find that day’s entry, the follow up gives the same error. The problem is thus in the Café.

but when I checked back the problem did not seem to exist any more. Strange.
- CommentRowNumber7.
- CommentAuthorzskoda
- CommentTimeFeb 17th 2021
- PermaLink
Author: zskoda
Format: MarkdownItexIt does again. Though the main page works.

It does again. Though the main page works.
- CommentRowNumber8.
- CommentAuthorTim_Porter
- CommentTimeFeb 17th 2021
- PermaLink
Author: Tim_Porter
Format: MarkdownItexThere is no problem for me at the moment.

There is no problem for me at the moment.

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nLab > Latest Changes: Ehresmann connection