Connection on a fiber bundle

(Ehresmann connection)

Definition and motivation

Let $E\to M$ be a fibre bundle, and let $V$ be the vertical bundle (subbundle of $TE$). We call a connection on $E$ to any subbundle (distribution) $H$ of $TE$ complementary to $V$ in the sense that

(and therefore $H$ is called a/the horizontal bundle).

The fundamental issue is that even though the vertical bundle is natural, as we have a horizontal projection onto $M$, there are many possible horizontal bundles, and one has to be fixed. In a trivial bundle $M\times F$, for example, we do have both projections, so we naturally have both vertical and horizontal sub-bundles. The resulting fibered connection is called a trivial connection.
If a bundle has a local trivialization in a neighborhood of $u\in E$ that sends the Ehresmann connection to the trivial connection, it is said that this connection is flat at $u$.

Tangent vectors mean directions. A vector in the vertical bundle specifies a vertical direction, that is to say, in the direction of the fiber. If you move like that, you don't leave the fiber. On the contrary, an horizontal vector will tell you how to travel from one fiber to another one. This brings us to the idea of parallel transport.

Connection 1-form

Other way to specify a connection consists, therefore, of a projection, i.e., a map $v:TE\mapsto TE$ such that $v^2=v$, and with $v(T_{u}E)=V_u$. In this case, $H_u=ker(v{|}_{T_{u}E})$. Remember that in any vector space $W$, a map such tat $p^2=p$ lets you decompose $W=ker(p)\oplus im(p)$ (see projection). The idea behind is that you can comprise $W$ to $p(W)$ in so many ways as horizontal subspaces choices.
So equivalently, the horizontal subbundle can be described by a $TE$-valued 1-form on $E$, called the connection form. It is the same as the structure 1-form of the underlying distribution on $E$.

If the bundle is a $G$-principal bundle then the connection could be a principal connection on a principal bundle an the connection 1-form would be a $\mathfrak{g}$-valued 1-form.

Point of view of jet bundle

In more technical terms, a connection is precisely a smooth section of the jet bundle

Particular case: vector bundles

(Wikipedia. See also this.)
Suppose that $E$ is a smooth vector bundle over $M$. Then an Ehresmann connection $H$ on $E$ is said to be a linear (Ehresmann) connection if $H_e$ depends linearly on $e\in E_x$ for each $x\in M$. To make this precise, let $S_{\lambda }$ denote scalar multiplication by $\lambda$ on $E$. Then $H$ is linear if and only if $H_{\lambda e}=\mathrm{d}(S_{\lambda }{)}_e(H_e)$ for any $e\in E$ and scalar $\lambda$.

Since $E$ is a vector bundle, its vertical bundle $V$ is isomorphic to ${\pi }^{\ast }E$. Therefore, if $s$ is a section of $E$, then $\mathrm{\Phi }(\mathrm{d}s):TM\to s^{\ast }V=s^{\ast }{\pi }^{\ast }E=E$. It is a vector bundle morphism and is therefore given by a section $\mathrm{\nabla }s$ of the vector bundle $\text{Hom}(TM,E)$. The fact that the Ehresmann connection is linear implies that, in addition, it verifies the Leibniz rule, i.e., $\mathrm{\nabla }(fs)=f\mathrm{\nabla }(s)+\mathrm{d}(f)\otimes s$, and therefore is a covariant derivative of $s$. That is, we have a vector bundle connection.

Conversely, a covariant derivative $\mathrm{\nabla }$ on a vector bundle defines a linear Ehresmann connection by defining $H_e$, for $e\in E$ with $x=\pi (e)$, to be the image $\mathrm{d}{s}_x(T_{x}M)$ where $s$ is a section of $E$ with $s(x)=e$ and ${\mathrm{\nabla }}_{X}s=0$ for all $X\in T_{x}M$.

Curvature

An important feature of a connection is the curvature.