How to relate the Ehresmann connection to connection 1-form?

[xXEhresmannXx]

I am studying connections on abstract manifolds. So far, I have read several equivalent definitions but I can't establish the equivalence between them on my own.

The first definition is the Ehresmann connection that defines a connection on a manifold as a distribution of vector spaces completing the vertical space in the tangent space of the total space at each point.

The second definition defines a connection on a manifold as a covariant derivative, i.e. a map

∇:Γ(E)→Γ(T∗M⊗E)

where π:E→M is a vector bundle and there is a version of the Leibnitz rule as follows:

∇X(fs)=df⊗s+f⋅∇Xs

for any section s and f∈C∞(M).

I tried to write things in a chart to find out how the covariant derivation is induced from a given connection but I couldn't proceed forward. I think I haven't understood the definitions well. Would someone clarify how a distributional connection give us a connection one-form and how we can recover the horizontal space if we have a connection one-form?

 

[Ben Niehoff]

An Ehresmann connection is more general than a connection on a vector bundle. It's probably more useful to understand vector bundles first.

Connections on vector bundles can be thought of as infinitesimal linear maps that relate the vector space at a point $x$ to the vector space at neighboring points $x + dx$.

Ehresmann connections are connections on general fiber bundles where the fiber is just some manifold, not necessarily with any sort of nice structure.

 

[strangerep]

The first definition is the Ehresmann connection that defines a connection on a manifold as a distribution of vector spaces completing the vertical space in the tangent space of the total space at each point.

Did it actually say "completing", or did it say "complementing"? I had understood an (abstract) Ehresmann connection to be essentially a smooth subbundle $H_M$ of $TTM$, called the horizontal bundle, which is complementary to the vertical bundle $V_M$ in $TTM$. I.e., $H_M \bigcap V_M = \{0\}$, and $TTM = H_M \oplus V_M$.

I tried to write things in a chart to find out how the covariant derivation is induced from a given connection but I couldn't proceed forward. I think I haven't understood the definitions well.

I know the feeling.

Would someone clarify how a distributional connection give us a connection one-form and how we can recover the horizontal space if we have a connection one-form?

I could maybe say a few things in terms of coordinates on the bundle (which mathematicians would probably dislike). Depends what you're really looking for.

 

[lavinia]

Would someone clarify how a distributional connection give us a connection one-form and how we can recover the horizontal space if we have a connection one-form?

- The horizontal space at a point is the kernel of the connection 1 form.

At each point in the fiber of a principal bundle, the tangent space is a direct sum of the vertical space and the horizontal space. The connection 1 form maps the tangent space into the Lie algebra of the the structure group. It does so by mapping the horizontal space to zero and the vertical space via the action of the structure group on the fiber.

Note that the connection 1 form is a Lie algebra valued 1 form not a number valued 1 form.

Also note the the invariance of the connection 1 form under the action of the structure group guarantees that the differential of the structure group preserves the horizontal spaces along each fiber.