What changes in parallel transport if we choose another connection (geometric intuition)?

Question

Since we can choose whichever connection we want, I wanted to know what changes in parallel transporting a vector if we choose two different connections for doing so(separately). A geometric explanation will be appreciated so I can visualize each parallel transport.
In addition, how can we compute or "see" the differences in each parallel transport?
Lastly, why should one choose the Levi-Civita connection apart from that it simplifies calculations(or it seems so due to the exchange symmetry in the two down indices(covariant) of the Christoffel symbols that correspond to the Levi-Civita connection--but, even if it turns out not to be the most simple connection to perform calculations with, the question still stands)?
Thank you.

Answer 1

The short answer to your first question is that if you change the connection, virtually anything can happen to parallel transport. More specifically, suppose $M$ is a smooth manifold, $\gamma\colon I\to M$ is a smooth embedded curve, $t_0\in I$, and $v$ is a vector in $T_{\gamma(t_0)}M$. If $V\colon I\to TM$ is any smooth vector field along $\gamma$ that satisfies $V(t_0) = v$, then we can find a connection on $M$ that makes $V$ into the parallel transport of $v$.

To see this, note first that we can find smooth vector fields $V_2,\dots,V_n$ along $\gamma$ so that $(V(t),V_1(t),\dots,V_n(t))$ forms a basis for $T_{\gamma(t)}M$ for each $t$. Then since $\gamma$ is an embedding, we can extend these vector fields to smooth vector fields on a neighborhood of the image of $\gamma$, and in some (possibly smaller) neighborhood they'll still be a local frame for $M$. In that neighborhood, we can define a connection by declaring these vector fields to be parallel (or, equivalently, by declaring the Christoffel symbols with respect to this frame all to be zeros). That connection can then be extended to all of $M$ using a partition of unity.

The best answer I know to your second and third questions is based on the following idea: If $M$ is a submanifold of $\mathbb R^n$ with its induced metric, and $V$ is a vector field tangent to $M$ along a curve $\gamma$ in $M$, then a natural and intuitive way to define what it means for $V$ to be parallel along $\gamma$ is to require that its ordinary directional derivative at each point of the curve be orthogonal to the tangent space of $M$ at that point. Intuitively, this means $V$ is "as close to constant as it can be" while staying tangent to $M$.

The point of the Levi-Civita connection is that it can be defined on an abstract Riemannian manifold in such a way that it's preserved by isometries, and if you happen to find an isometric embedding of your manifold into Euclidean space (which always exists by the Nash embedding theorem), then parallelism with respect to the Levi-Civita connection corresponds to the intuitive notion described above.