Tangent bundle

  • Thread starter math6
  • Start date
  • #1
67
0
hello friends
my question is: if we have M a compact manifold, do we have there necessarily TM compact ?
thnx .
 

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
Gold Member
4,783
18
Of course not. Just loot at one tangent space. On the one hand, that's closed in TM, and on the other hand its homeomorphic to R^n (not compact). So TM cannot be compact, otherwise each tangent space would be too.
 
  • #3
mathwonk
Science Advisor
Homework Helper
2020 Award
11,153
1,347
one way to make it compact is to replace TX by PTX, the bundle whose fibers are the projective spaces associated to the tangent vector spaces.
 
  • #4
67
0
thnx for answers are you sure mathwonk for the answers can you give me proof if you can please ?
 
  • #5
662
1
one way to make it compact is to replace TX by PTX, the bundle whose fibers are the projective spaces associated to the tangent vector spaces.

Aren't those the tautological bundles.?
 
  • #6
662
1
Never mind, Wonk, I spoke too soon, there is just a vague relation.
 
  • #7
662
1
Math6:
I am not sure I understood your question, but Projective spaces are compact
because they are the continuous image ( under the quotient map) of the
compact space S^n, and so they are compact.
 

Related Threads on Tangent bundle

  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
8
Views
11K
Replies
5
Views
7K
Replies
7
Views
4K
  • Last Post
Replies
2
Views
2K
Replies
11
Views
2K
Replies
4
Views
4K
  • Last Post
Replies
3
Views
4K
Replies
55
Views
5K
Top