Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

On the page 32 of Lee's book Manifolds and differential geometry, he writes:

In the definition of path connectedness..., we used continuous paths, but it is not hard to show that if two points on a smooth manifold can be connected by a continuous path, then they can be connected by a smooth path.

I do not quite understand.

First of all, what does a path being smooth mean? A path on a manifold is a function: $$ p:[0,1]\rightarrow M $$

I think to be continuous is in the topological sense.

But what does smooth mean? Is it in the diffeomorphic sense?

Do we regard $[0,1]$ as a set in the manifold $\mathbb{R}$ and applying the definition of smoothness of function between manifolds?

After all, I wonder how to prove this statement.

share|improve this question
1  
In down-to-earth terms it means that the derivative $p^\prime(t)$ exists at every $t\in(0,1)$. Then, the claim is that every continouous path in a smooth manifold can be deformed to a smooth map. This results from a Smoothing Lemma which is usually given for granted in any graduate course on the subject ... :) –  Andrea Mori Oct 22 '12 at 11:15
    
..all right.. I will take it for granted too..:) –  hxhxhx88 Oct 22 '12 at 12:04

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.