Complete vector field X => X defined on the whole manifold?

[center o bass]

According to Isham (Differential Geometry for Physics) at page 115 he claims:

"If X is a complete vector field then V can always be chosen to be the entire manifold M"

where V is an open subset of a manifold M. He leaves this claim unproved.
A complete vector field is a vector field which has integral curves defined on the whole of $\mathbb{R}$. Does the claim somehow follow from the definition?

 

[pasmith]

According to Isham (Differential Geometry for Physics) at page 115 he claims:

"If X is a complete vector field then V can always be chosen to be the entire manifold M"

where V is an open subset of a manifold M. He leaves this claim unproved.
A complete vector field is a vector field which has integral curves defined on the whole of $\mathbb{R}$. Does the claim somehow follow from the definition?

How is V defined? M is always an open subset of itself, but without knowing how V relates to X we really can't help you.

 

[center o bass]

How is V defined? M is always an open subset of itself, but without knowing how V relates to X we really can't help you.

Isham defines it as follows:

"Let X be a vector field defined on an open subset U of a manifold M and let p be a point in U c M . Then a local flow of X at p is a local one-parameter group of local diffeomorphisms defined on some open subset V of U such that p $\in$ V c U and such that the vector field induced by this family is equal to the given field X ."