# Condition on vector field to be a diffeomorphism.

• I
• kroni

#### kroni

Hi everybody,

Let $V(x)$ a vector field on a manifold ($R^2$ in my case), i am looking for a condition on $V(x)$ for which the function $x^µ \rightarrow x^µ + V^µ(x)$ is a diffeomorphism. I read some document speaking about the flow, integral curve for ODE solving but i fail to find a generic condition to avoid V to send two point on the same coordinate. I think about the generator of the diffeomorphism group but it's only defined infinitesimaly.

Thanks

Clément

there are a lot of possible approaches to sufficient conditions. For example, if ##\sup_{x\in\mathbb{R}^2}\Big\|\frac{\partial V}{\partial x}\Big\|## is small enough then it is a diffeomorphism. monotonicity assumption can also help. Perhaps the book Topics in Nonlinear Functional Analysis by L Nirenberg would be of use

Monotonicity work only in 1D,
[itex]\sup_{x\in\mathbb{R}^2}\Big\|\frac{\partial V}{\partial x}\Big\|[\itex] is non local.
I will look in the book you advise. I find this problem really interesting, may be treated and treated again, but interesting