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I am currently studying a first course in manifolds and I have come across a definition that I don't really understand:

Let $X$ be a vector field on a manifold $\mathcal{M}$. If all the integral curves of $X$ extend $\forall t\in\mathbb{R}$, then we say that $X$ is complete.

So my intuition for what this is saying is as follows: A vector field $X$ is complete if $\forall p\in\mathcal{M}$ and any given integral curve $$ \gamma :(a,b)\to\mathcal{M} $$ passing through the point $p$, the curve $$ \gamma : \mathbb{R} \to\mathcal{M} $$ is an integral curve of $X$ passing through $p$.

Is this definition equivalent to the one given in my lecture? I just felt like the original definition lacked enough rigour for my liking an my lecturer skipped past this definition without really explaining it. Also could anyone give any examples or counterexamples of complete vector fields to ground my understanding? Thanks in advance!

user320140
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    Your definition is not so formal. When you say "the curve $ \gamma:\mathbb{R}\to \mathcal{M}$ is an integral curve of $X$ trough $p$, what is $\gamma$? You should instead write that $\gamma:(a,b)\to \mathcal{M}$ can be extended to an integral curve of $X$ defined over the whole $\mathbb{R}$. – Overflowian Oct 30 '17 at 09:16
  • Yeah sure, here we define an integral curve of $X$ through $p$ as a path $\gamma:(a,b)\to\mathbb{R}$ such that $\gamma(o) = p$ (we assume that $0\in(a,b)$ here) and such that $$(\gamma){*} = X{\gamma(t)}, $$ where $(\gamma)_{*}$ is a push-forward. I mean my aim here isn't to produce a completely rigouros definition but just to make sure that I do understand the definition given in the lecture. – user320140 Oct 30 '17 at 09:44

1 Answers1

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As mentioned in the comments, it's better to say you can extend $\gamma: (a,b) \to \mathcal{M}$ to an integral curve of $X$ defined on $\mathbb{R}$. In this wording it is indeed equivalent to the definition given in the lecture. Another equivalent definition would be "$X$ is a complete vector field if and only if it has a global flow".

Here are also some nice examples of vector fields on $\mathcal{M} = \mathbb{R}^2$.

$$ X_1 = \frac{\partial}{\partial x_1} \\ X_2 = x_1 \frac{\partial}{\partial x_1} + x_2 \frac{\partial}{\partial x_2} \\ X_3 = x_2 \frac{\partial}{\partial x_1} - x_1 \frac{\partial}{\partial x_2} $$ It isn't hard to find the corresponding integral curves and thus proving they are complete. An easy example of a vector field that isn't complete would be $X = \frac{\partial}{\partial x_1}$ on the manifold $\mathcal{M} = \mathbb{R}^2 \setminus \{0\}$.

Demophilus
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  • Great this helps a lot. I have literally just covered what a flow is so I'm not entirely sure what is meant by a global flow yet. Thank you for the examples and counter-example also. – user320140 Oct 30 '17 at 09:49