# Flows on a manifold as a group action.

A flow on a manifold $M$ is a one-parameter group of diffeomorphism $\psi_t: M \to M$. In particular, this induces a smooth group action on $M$, $$\mathbb{R} \times M \to M$$ $$(t,m) \mapsto \psi_t(m)$$ So that we have a group homomorphism $\phi: \mathbb{R} \to Diff(M)$.

My question is the following: Given any diffeomorphism $g \in Diff(M)$, does there exist a flow (and thus a vector field) on $M$ such that $g$ is in the image of $\phi$?

• I added the "differential-topology" tag to your post. Cheers! – Robert Lewis Apr 22 '18 at 5:04

The answer is no, and the reason may be concisely stated: for any flow

$\phi: \Bbb R \times M \to M \tag 1$

each

$\phi_t: M \to M, \; \forall x \in M \; \phi_t(x) = \phi(t, x) \tag 2$

is homotopic to the identity map, via a homotopy

$H(s, t, x) = \phi_{st}(x), \; s \in [0, 1]; \tag 3$

it is easily seen that

$H(0, t, x) = \phi_0(x) = x, \; H(1, t, x) = \phi_t(x). \tag 4$

For many $M$ there are, however, diffeomorphisms $\psi:M \to M$ which are not homomtopic to the identity; an example is provided by the antipodal map $\psi(x) = -x$ on $S^{2n}$ for any positive integer $n$.

No, the group of diffeomorphism of a manifold is not always connected, for example for the torus, the connected components are in bijection with $SL(2,\mathbb{Z})$.

https://en.m.wikipedia.org/wiki/Mapping_class_group_of_a_surface

• Ok. So we definitely would need a manifold to be connected for this to hold. Does it necessarily hold for, let us say, compact, connected manifolds? – Jadwiga Apr 22 '18 at 5:02
• The two dimensional torus is connected. – Tsemo Aristide Apr 22 '18 at 5:03