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This question is very closely related to another one I just asked. The general question is to what extent there is a theory of differential equations for smooth correspondences (between a smooth manifold and $\mathbb R$, say). But I'm wondering about a specific well-studied case.

This question is very closely related to another one I just asked. The general question is to what extent there is a theory of differential equations for smooth correspondences (between a smooth manifold and $\mathbb R$, say). But I'm wondering about a specific well-studied case.

(If $N$ is not compact, then the flow map does not have global-time solutions. But I can still do everything; I just have to replace the space $TN \times \mathbb R$ by an open subspace. In fact, $\phi$ still defines on $TN \mathbb R$ the structure of an action groupoid, and both $(\pi, \pi\circ \phi, t)$ and $S$ are groupoid homomorphisms, so that the above correspondence is a span of groupoids. Does this enter the discussion in any interesting way?)

(If $N$ is not compact, then the flow map does not have global-time solutions. But I can still do everything; I just have to replace the space $TN \times \mathbb R$ by an open subspace. In fact, $\phi$ still defines on $TN \times \mathbb R$ the structure of an action groupoid, and both $(\pi, \pi\circ \phi, t)$ and $S$ are groupoid homomorphisms, so that the above correspondence is a span of groupoids. Does this enter the discussion in any interesting way?)