Canonical notion of parallel transport

Question

I have a "What is the right search term?" style question:

Suppose $S\subset\mathbb{R}^3$ is a surface and that we are given two points $x,y\in S$. Furthermore, take $v_x\in T_x S$ to be a tangent vector at $x$.

Is there a canonical notion of parallel transport that gives a vector $v_y\in T_y S$ "corresponding to" $v_x\in T_x S$ without choosing a path from $x$ to $y$?

That is, I want a way to make tangent spaces at $x,y\in S$ "talk to each other" without necessarily choosing a path between them (or at least having a good justification for the path I choose).

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Of course, we could do something like consider parallel transport along the geodesic from $x$ to $y$. But it seems like this could have regularity issues at the cut locus. I'm hoping that there is some classical, well-studied construction that I failed to notice/find. The Sasaki metric on the tangent bundle seems relevant, but I will admit I had trouble following the literature in this area.

Answer 1

It's risky to give a blanket negative answer to so open-ended a question (particularly, one in which "canonical" is undefined), but with minor qualifications the answer appears to be "no". Here are a few easy observations, a bit too long for a comment:

1. If for all $x$ and $y$ in $S$, there is a canonical parallel transport between $T_{x}S$ and $T_{y}S$, then (unless the definition of canonical permits the parallel transport of a frame to be discontinuous) $TS$ is topologically trivial.

2. If $S$ is compact, then for some $x$ and $y$ in $S$, there exist geodesics $\gamma_{1}$ and $\gamma_{2}$ of equal length between $x$ and $y$ along which parallel transport differs. (I don't have a completely rigorous proof, but here's a sketch: Pick an arbitrary point $x_{0}$ of $S$ and non-zero tangent vector at $x_{0}$. The geodesic starting at $(x_{0}, v) \in TS$ must cross itself after finite distance (because $S$ is not flat), and for generic initial conditions the geodesic is not closed, i.e., when it crosses itself at a point $x$, the two velocities are unequal. Now fix a "loop" of geodesic starting at $x$, and let $y$ be the point halfway along the loop.)

On the positive side (with substantial restrictions on $S$):

3. If any two points of $S$ are joined by a unique minimizing geodesic (e.g., $S$ is isometric to a geodesically-convex subset of the hyperbolic plane, or an open round hemisphere), you can parallel transport uniquely along the minimizing geodesic from $x$ to $y$.

4. If $S$ is connected, simply-connected, and flat, then parallel transport is path-independent because the holonomy is trivial. (Connected and flat are not enough. (!) Think of a cone with its vertex removed.)

Answer 2

Been a long time Justin! I don't know the literature, but here is an approach towards formulating a practical relaxation of the problem.

If you relax the requirements of parallel transport a bit, you might be able to get something reasonable by formulating this as a regularized optimization problem over the whole tangent bundle at once.

Imagine discretizing your manifold, and at each discrete cell $i$ attaching two tangent vectors $u_i, v_i$ that are supposed to serve as an "orthonormal" basis for the tangent space there. Now use springs to connect the tip of vector $u_i$ to the corresponding tips of vectors in neighboring cells $u_j,~ j ~\text{neighbor of}~ i$, and then similarly connect up the vectors $v_i$ with neighbors. If the discretization is fine enough, you can allow the springs to pass through the ambient space in straight lines; otherwise you would need to constrain them to be geodesics.

Then you can solve for the configuration minimizing the total spring energy, with a regularization that penalizes non-orthonormality of basis vectors at each point. You will also probably need to fix the basis on a particular cell in order rule out the null space associated with rotations in all tangent spaces simultaneously. In particular, let

• $p_i$ be the base point associated with cell $i$,
• $B_i$ be some fixed basis for the tangent space at point $i$,
• $u_i, v_i$ be the approximately orthonormal vectors we are looking for that change smoothly from point to point and lie in the tangent space,
• $x_i, y_i$ be the coefficients of $u_i, v_i$ in the basis $B_i$. I.e., $u_i = B_i x_i$, $v_i = B_i y_i$.
• $X, Y$ be the matrices with vectors $x_i, y_i$ as columns, respectively.

Then we consider the problem \begin{align} \min_{X, Y}~& \frac{1}{2}\sum_{i~\text{neighbor of}~j}||(p_i + B_i x_i) - (p_j + B_j x_j)||^2 + ||(p_i + B_i y_i) - (p_j + B_j y_j)||^2 \\ &+ \frac{\alpha}{2}\sum_i||\begin{bmatrix}x_i & y_i\end{bmatrix}^T \begin{bmatrix}x_i & y_i\end{bmatrix} - I||_\text{Fro}^2 \\ &+ \frac{\beta}{2}||x_0 - x_0^\text{fix}||^2 + \frac{\beta}{2}||y_0 - y_0^\text{fix}||^2. \end{align}

Once this is solved, you can "transport" a tangent vector $w$ at cell $i$ to a far away cell $k$ by expressing $w$ in terms of the local basis $u_i, v_i$, then using the coefficients in that basis to form a linear combination of the far away basis vectors $u_k, v_k$. I.e., if $$w = c_u u_i + c_v v_i$$ then $$T_{i\rightarrow k} w := c_uu_k + c_v v_k.$$

All terms in the optimization objective will be very nice, so you can solve it with fast trust region Newton-type methods. With some work this optimization problem can probably even be formalized in terms of function spaces defined on the manifold, then discretized in a more careful manner respecting the manifold structure.

I've used some similar ideas before to interpolate between orthonormal frames in high dimensions, and have some ideas about preconditioning the hessian for this sort of problem - you have my email; let me know if you want to discuss further.