Does a Compact Manifold Imply a Compact Tangent Bundle?

[math6]

hello friends
my question is: if we have M a compact manifold, do we have there necessarily TM compact ?
thnx .

 

[quasar987]

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.

 

[mathwonk]

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.

 

[math6]

thnx for answers are you sure mathwonk for the answers can you give me proof if you can please ?

 

[Bacle]

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.?

 

[Bacle]

Never mind, Wonk, I spoke too soon, there is just a vague relation.

 

[Bacle]

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.