# A problem about integral curves on a manifold

1. Nov 8, 2008

### quasar987

I must demonstrate in two ways that if c(t) is an integral curve of a smooth vector field X on a smooth manifold M with c'(t_0)=0 for some t_0, then c is a constant curve.

I found one way: If $\theta$ denotes the flow of X, then because X is invariant under its own flow, we have

$$c'(t)=X_{c(t)} = (\theta_{t-t_0})_*X_{c(t_0)}=(\theta_{t-t_0})_*c'(t_0)=0$$

for all t.

Does anyone see another way?

2. Nov 10, 2008

### quasar987

(Solved. I had initially thought that and argument based directly on the fundamental theorem on EDO does not work, but it does.)