Given a smooth manifold $M$, a smooth vector field $V$, and some $x_0 \in M$, we can define the exponential map through the map $\phi$, where $\phi$ is defined on some open interval of $\mathbb{R}$, $\phi(t_0) = x_0$, and $x = \phi(t)$ satisfies the differential equation
$$\frac{dx}{dt} = V(x),$$
and we let $\phi(t) = \exp(t\cdot V, x_0)$, so $\exp(v, x) = \phi(1)$.
This is taken from Theorem 3.1 of J-M Souriau's Structure of Dynamical Systems. At this point of the book, he hasn't introduced any concept of an affine connection, only assuming a Hausdorff manifold, but here we have somehow defined (and shown the existence of a unique) exponential map. Is there some natural affine connection he has assumed, or can we find the connection corresponding to this given definition of an exponential map? I saw this related question: Exponential maps depends on Riemannian metric?, but I'd like to pinpoint where exactly the metric or associated connection would show up.
I know that the exponential map can be defined in terms of geodesics, and I guess the ODE $dx/dt = V(x)$ is the geodesic equation, which can be solved componentwise with respect to some a suitable coordinate basis on the tangent bundle (e.g., Prop 6.1.2 here). But what metric or connection does this geodesic equation correspond to?
Best Answer
The author seems to refer to exponential map for any "time 1 of a flow". This is a sort of generalization of:
These notions are the same in some particular cases - when a Lie group is equipped with a bi-invariant metric. In general, they differ, but they share some common properties, for example, $\psi_t\psi_s = \psi_{t+s}$ and $\psi_0=\mathrm{id}$, which is a very-well known property of the complex exponential function $\exp : \mathbb{C}\to \mathbb{C}^*$.
There is no reason for a time 1 map of a flow to be induced by some geodesic flow of a particular riemannian metric, so there may be no connextion inducing this flow and giving sense to the term "exponential map" as understood as a riemannian notion.