- #1

- 7,084

- 10,725

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.