Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?

Question

Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to M$ that sends a tangent vector $v \in T_p M$ to the endpoint of the unique geodesic $\gamma$ satisfying $\gamma(0)=p$ and $\gamma'(0)=v$.

In particular, the notion of a geodesic allows us to define an exponential map. My question concerns the reversal of this construction:

Suppose that for each $p \in M$, we have a map $\sigma_p : T_p M \to M$ with some desirable properties. Can one define a notion of $\sigma$-geodesic that coincides with notion of a geodesic when $\sigma =\exp$?

If this can be done, I suspect that it is well-known, but I've never come across such a construction.

Answer 1

Exponential map in your definition is closely related to the smooth family of smooth curves smoothly depending on the position such that in every point in every direction there exists precisely one curve at this point in this direction. In literature, such a family is sometimes called path structure.

There are two differences between your structure and path structure: the first one is that you curves are parameterised, and the curves of path structure are not. But this an additional structure, so my answer below will has sense also for your setup.

The other difference is that for a path structure there exists precisely one, or, in the case of irreversible path structures, at most two curves from the family. You definition allows that (for three points A,B,C) the curves from A and B to C come to C with the same velocity vector which is not possible for geodesics. Or, even more wild, take 3 points A, B, C and consider the trajectory $\gamma_{ABC} $ of your exponential map which starts from A and then passes through first B and then C (assume such trajectory exists, i.e., choose B and C on a trajectory from A). Your definition does not require that the segment of $\gamma_{ABC}$ from $B$ to $C$ is a trajectory of your exponential map starting from $B$ and going to $C$, though this property may be essential for geodesics.

But, returning to path structures, most path structures do not come from an affine connection (known since at least Cartan for dim 2, essentially the same proof works for all dimensions, see https://arxiv.org/abs/1101.2069). In dimension 2, every reversible path structure is locally Finsler-metrisable https://arxiv.org/abs/1002.0243, the question whether every irreversible path structures is metrisable is still open.
In higher dimensions most path structures are, even microlocally, not Finsler-metrisable, see (the open access paper) https://link.springer.com/article/10.1007/s10714-022-03006-2 .

Answer 2

Let’s define $\gamma:\mathbb{R}\to M$ as a $\sigma$-geodesic iff for any $a,b,c\in\mathbb{R}$, there are vectors $v,w$ in the tangent space at $\gamma(a)$ with $$\sigma_{\gamma(a)}(v)=\gamma(b)$$ $$\sigma_{\gamma(a)}(w)=\gamma(c)$$ $$(c-a)v=(b-a)w$$

If $\sigma=\exp$, and $\gamma$ is a geodesic, this property clearly holds.

If $\sigma=\exp$, and this property holds for $\gamma$, then let $P(a,k)$ be the statement that $d(\gamma(a),\gamma(b))=k|a-b|$ whenever $\gamma([a,b])$ is entirely within the injectivity radius of $\gamma(a)$. Then there is some $k$ with $P(0,k)$, and for that $k$, the set $\{a:P(a,k)\}$ is both open and closed in $\mathbb{R}$. So in fact $P(a,k)$ holds for all $a$, $d(\gamma(a),\gamma(b))=k|a-b|$ for all sufficiently close $a$ and $b$, and therefore $\gamma$ is a geodesic.