Does a Compact Manifold Imply a Compact Tangent Bundle?

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Discussion Overview

The discussion centers on the relationship between a compact manifold and the compactness of its tangent bundle. Participants explore whether the tangent bundle of a compact manifold must also be compact, examining various mathematical properties and examples.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant asserts that the tangent bundle TM cannot be compact because individual tangent spaces are homeomorphic to R^n, which is not compact.
  • Another participant suggests that one way to achieve compactness is to consider the projective tangent bundle PTX, which consists of projective spaces associated with the tangent vector spaces.
  • A later reply questions the relationship between the projective spaces and the tangent bundle, indicating some uncertainty about the implications.
  • Another participant notes that projective spaces are compact due to being continuous images of compact spaces, specifically S^n.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the tangent bundle of a compact manifold is compact. Multiple competing views are presented, with some arguing against compactness and others proposing alternative constructions that may lead to compactness.

Contextual Notes

There are unresolved mathematical steps regarding the implications of the properties of tangent spaces and projective spaces. The discussion also reflects varying levels of understanding among participants regarding the definitions and relationships involved.

math6
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hello friends
my question is: if we have M a compact manifold, do we have there necessarily TM compact ?
thnx .
 
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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.
 
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.
 
thnx for answers are you sure mathwonk for the answers can you give me proof if you can please ?
 
mathwonk said:
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.?
 
Never mind, Wonk, I spoke too soon, there is just a vague relation.
 
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.
 

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