Defining the exponential map without explicit affine connection

Question

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?

Answer 1

He should not have called this map exponential. The most common name for it is the "time 1 flow of the vector field $V$."

Here is how this relates to the exponential map in differential geometry. What you need for exponential map is an affine connection $\nabla$ on the tangent bundle $TM$, not a Riemannian metric. (Although, a Riemannian metric defines the Levi-Civita connection that you can then use.) The connection $\nabla$ defines a certain vector field $W$ on $TM$. The time one flow $F$ of $W$ is closely related to the exponential map of $M$: Given a point $p\in M$ and an tangent vector $v\in T_pM$, consider the vector $W(v)\in T_vTM$. Then $F(W(v))=(p',v')$, where $p'\in M$ and $v'\in T_{p'}M$. Lastly, $\exp_p(v)=p'$.

My favorite reference for this is "Riemannian Geometry" by do Carmo.

Answer 2

The author seems to refer to exponential map for any "time 1 of a flow". This is a sort of generalization of:

• the exponential map of a Riemannian manifold, which is in fact the time 1 map of the geodesic flow (defined on the unit tangent bundle)
• the exponential map of a Lie group, which is the time 1 map of a natural flow of left invariant vector fields

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.