General Doubts on Ehresmann Connections

[WWGD]

Hope someone can help me clarify a couple of issues about
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.

 

[lavinia]

As I understand it. the Ehresmann connection is defined on a smooth fiber bundle so the base and the total space are manifolds. Since the projection map from the total space to the base is smooth and of maximal rank, each fiber is a manifold by the Implicit Function Theorem.

Horizontal and vertical subspace are subspaces of the tangent spaces to the total space of the bundle. This means you are talking about manifolds.

Riemannian metrics are defined on vector bundles. They may or may not be smooth.

There is work on the analogue of bundles in the PL category. Don't know anything about it.

 

[WWGD]

Thanks, lavinia, that helps clear things out.

I was hoping you or someone else could suggest a book that
does actual calculations of this sort, i.e., not a purely-abstract treatment,
and calculate too, e.g., Lie derivatives, connections. I'm interested in the actual
derivation too, but it helps me to see calculations done too.
Anyone, please?

 

[lavinia]

Thanks, lavinia, that helps clear things out.

I was hoping you or someone else could suggest a book that
does actual calculations of this sort, i.e., not a purely-abstract treatment,
and calculate too, e.g., Lie derivatives, connections. I'm interested in the actual
derivation too, but it helps me to see calculations done too.
Anyone, please?

I couldn't agree with you more that calculations are necessary for learning this stuff.

I only know about connections on vector bundles and on principal fiber bundles. Spivak's second volume on differential geometry is devoted to these. I find the book difficult going because there is too much computation but he does try to be conceptual as well. There is also Sternberg's Curvature in Mathematics and Physics. Principal G-bundles are key to modern Differential Geometry and Physics e.g. Gauge Theory. You need to learn some things about Lie groups for this and these have many nice calculations that are easy to understand.

For a simple version to get you started there is Singer and Thorpe's wonderful book Lecture Notes on elementary Geometry and Topology. They treat Levi-Civita connections on the tangent bundle to surfaces as connections on principal SO(2) bundles. All of the ideas are laid out in this simple situation.

I am also willing to work with you on this since I am still learning it. A good start would be to go through the Levi Civita connection on the bundle of oriented frames tangent to a manifold embedded in Euclidean space.

To get you started notice that for an embedded manifold a frame (e1,...en) is a vector of smooth maps of the frame bundle into Euclidean space.

The differential forms

<dei,ej>

are the components of the connection 1 form which takes values in the Lie algebra of SO(n). Notice that the matrix of these forms is skew symmetric.

 

[blue_raver22]

I can understand your confusion and doubts about Ehresmann connections and vertical spaces in fiber bundles. These concepts can be quite abstract and difficult to grasp, especially when dealing with general fibers that are not manifolds or vector spaces.

To answer your first question, the tangent space at a point in a general fiber can be defined by considering the fiber as a local product space. This means that we can decompose elements in local trivializations and use this decomposition to define the tangent space. This approach can be applied to any type of fiber, not just manifolds or vector spaces.

For your second question, in order to define a Riemann metric over a general space E, 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}. This normal space can then be used to construct a Riemannian metric over E. This approach allows for a generalization of the concept of a metric to any topological space, not just manifolds.

I hope this helps clarify some of your doubts about Ehresmann connections and vertical spaces. Keep exploring and asking questions, as understanding these concepts is crucial for studying fiber bundles and their applications in various areas of mathematics and physics.