4

I have a stupid and probably naive question about one line in the book of Milnor about Morse theory. What does exactly means if $v \in T_pM$ then there is an associated vector field $\tilde v $ ?

I have a kind of vague idea of what could be this vector fields (identify $T_pM$ and $T_qM$ for $p,q$ close and show that this does not depends of this identification, and that we can extend this to all $M$).

enter image description here

user48483
  • 43
  • 2
    It may help if you attach some copy of the relevant page. However, my guess, without having read the book, is that $\tilde{v}$ is a vector field such that $\tilde{v}(p)=v$. – Amitai Yuval Jun 22 '15 at 22:35
  • so is it simply the constant vector field ? – user48483 Jun 22 '15 at 22:40
  • @AmitaiYuval : I'm sorry I need 10 reputation to add picture. But I think you're right – user48483 Jun 22 '15 at 22:44
  • 1
    Oh, I didn't know about the reputation issue. Note that in general there isn't such a thing as "constant vector field", since there is no canonical way to identify $T_pM$ with $T_qM$ for $p\neq q$. – Amitai Yuval Jun 22 '15 at 22:46
  • Thanks for the useful precisions. Also, (last question, I hope I'm not taking your time too much) Milnor is talking about "poisson bracket", what is this operator (in this particular case) ? (I have 10 now I will try to put the image again) – user48483 Jun 22 '15 at 22:48
  • @AmitaiYuval ...and for be honnest I don't really understand exactly what happened, the notation make me feel a bit lost. Do you know something I can read quickly for feel more confortable with this ? (Maybe I simply need to read more before reading this book of Milnor !) And sorry, if I'm asking too much just simply don't answer :) – user48483 Jun 22 '15 at 22:52
  • The only Poisson bracket I know exists only on symplectic manifolds. Differential geometry, like anything else in mathematics, takes its time to digest. Take it easy. – Amitai Yuval Jun 22 '15 at 22:53
  • Ok, thanks again for the useful precisions ! – user48483 Jun 22 '15 at 22:54
  • OK, now that I read the page, I can tell you for sure - this shouldn't be the first book you read on manifolds. – Amitai Yuval Jun 22 '15 at 22:57
  • I had one course in algebraic topology + the book of Lee on topological manifold. I feel it's not enough, I will read this later, thanks again for your patience. – user48483 Jun 22 '15 at 23:00

1 Answers1

2

Milnor just means that $\tilde{v}$ is any vector field such that $\tilde{v}_p=v$. Any $v\in T_p M$ can be extended to a vector field. See Lee, Smooth Manifolds, page 84, Lemma 4.5 for a proof.

Also, where he says "Poisson bracket" he meant to say "Lie bracket".

Seth
  • 9,393
  • 1
  • 27
  • 64