Hope someone can help me clarify a couple of issues about(adsbygoogle = window.adsbygoogle || []).push({});

Ehresmann connections and vertical spaces. My main issue is how to define

the tangent space at a point in a _general_ fiber F in a fiber bundle (and

not just when F is either a vector or a manifold If I understood correctly) when finding the vertical and horizontal spaces on the top space E of a general bundle (E,B, pi,F) . More specifically , this is the way I understand it :

1) We want to find the tangent space at a point

of a fiber, where the fiber is not a manifold,in order to define the

vertical space T_eE at a point e in E: .

The vertical space T_eE at a point e in E , of a general bundle (E,B,pi, F) ;

E the total space; B the base space ;pi the projection map and F the fiber ,

is defined by:

1.1) Project e down to B ,i.e., pi(e)=x

1.2) Now lift x back up ,so that pi^{-1}(x):=F_x ; _Fx is the fiber over x.

--------------------------------------------------------------------

1.3)T_e F_x is the vertical space of e in E .

** QUESTION ** How does one define the tangent space over a general fiber,

; specifically,when the fiber is neither a manifold nor a vector space.

Do we decompose (local) product spaces, i.e., do we decompose elements in

local trivializations?

-----------------------------------------------------------------------------------

2) There is a general result that every bundle (E,B,pi,F) as above,

can be given an Ehresmann connection. Now,let's assume we have found a

way of defining a tangent space T_e Fx at the general fibers Fx

All the answers I know for doing this say that (paraphrase):we give E a Riemann

metric <,>( so that we can use the inner-product to define a normal space

N as N:={ n in E: <n,v>=0 for v in T_eFx}.)

***Question ** How does one define a Riemann metric over a general space

E,when E is not necessarily a manifold? A metric is defined, AFAIK,as

at the level of the tangent bundle,i.e.,as a (2,0) tensor field.

How is this generalized when E is a general topological space,with no

"natural" definition of tangent spaces,let alone tangent bundles ?

I hope I'm not too far off.

Thanks in Advance.

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# General Doubts on Ehresmann Connections

Loading...

Similar Threads - General Doubts Ehresmann | Date |
---|---|

I About extrinsic curvature | Jul 16, 2017 |

A Diffeomorphisms & the Lie derivative | Jan 23, 2017 |

A Two cones connected at their vertices do not form a manifold | Jan 10, 2017 |

A Notation in Ricci form | Oct 17, 2016 |

How to relate the Ehresmann connection to connection 1-form? | Aug 24, 2015 |

**Physics Forums - The Fusion of Science and Community**