Understanding the choice in the horizontal subspace

Question

I am taking a course in differential geometry but my background is in physics. I am struggling to understand how there is an 'choice' in the horizontal subspace when one is considering a vector bundle $\pi : E \rightarrow B$

The vertical subspace at a point $p \in E$ is defined as the vector space $ker(d\pi)_p$. Call this $T_{v_p}E$ . A horizontal subspace $S_p$ is defined as being such that $S_p \cap T_{v_p}E = \{ 0 \}$ and $S_p \bigoplus T_{v_p}E = T_pE$

Now perhaps I am misunderstanding how the direct sum works, but it seems to me that $S_p$ is just the complement of $T_{v_p}E$ in $T_pE$. Is the complement not just the entire rest of $T_pE$ (and add the zero vector)? So where is the 'choice'?

One possibility I have thought of is: considering that the Cartesian plane $\mathbb{R}^2 = \mathbb{R}\bigoplus \mathbb{R}$, and supposing $T_{v_p}E$ were the line $x+y = 0$, the the horizontal subspace could be $y-x=a$ for any $a\in \mathbb{R}$. Are these different spaces? Is this what we mean by the choice in horiztonal subspace?

Also, I have read that $S_p$ is isomorphic to $T_pE/T_{v_p}E$ (i.e. the quotient). Is quotienting vetor spaces undoing a direct sum?

Edit: Adding an initial thought on my example with $\mathbb{R}^2$, only one of the lines passes through the zero vector and thus is a proper subspace. So this cannot be correct.

Answer 1

The "entire rest" means little. Suppose $V$ is a vector space, $U$ a proper subspace. You can define the complement to be $V/U$, and you can give it the structure of a vector space, but it is not a vector subspace of $V$.

A complementary subspace $W$ is a vector subspace of $V$ such that $U\cap W= \{0\}$ , $U+W=V$. However if $V$ comes with no additional structure there are plenty of ways (infinitely many) to choose $W$. For a simple example take $V=\mathbb{R}^2$, $U=\mathbb{R}(1,0)$, the 1d vector space generated by the vector $(1,0)$. Then any vector space $\mathbb{R}(a,b)$, $b\neq 0$, that is any line through the origin which is not the $x$-axis, satisfies the required properties.

Therefore, without additional structure, the vertical subspace at a point of a vector bundle is canonically given (i.e. it involves no arbitrary choices), the horizontal one is not. A choice of a horizontal subspace with the required properties is an additional structure known as a connection.

If the vector space has some additional structure, a horizontal space may be canonically given. Notably, if $V$ has an inner product then the orthogonal complement of any vector subspace is uniquely defined, and this gives you a canonical way of defining the horizontal subspace.

This corresponds to the fact that if you are given a manifold embedded in Euclidean space you recover the covariant derivative of a vector field with respect to the Levi-Civita connection by taking the ordinary derivative and projecting (via the Euclidean metric) orthogonally onto the tangent space of the manifold (this last remark assumes various things, for example the relation between a horizontal distribution and a covariant derivative, which are quite non-trivial, so don't worry if you don't follow the details).