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I have a question about the compactness of the tangent bundle: assume that the manifold M is compact, does it make necessarily TM compact ? if not TM, a submanifold of TM (precisely a submanifold of vector norm equal to 1) can be compact?

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I have a question about the compactness of the tangent bundle: assume that the manifold M is compact, does it make necessarily TM compact ? if not TM, a submanifold of TM (precisely a submanifold of vector norm equal to 1) can be compact?

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quasar987

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I'm not sure what you're asking in your second question.

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lavinia

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I have a question about the compactness of the tangent bundle: assume that the manifold M is compact, does it make necessarily TM compact ? if not TM, a submanifold of TM (precisely a submanifold of vector norm equal to 1) can be compact?

The tangent unit sphere bundle of a compact manifold is compact. It is not hard to prove this.

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thnx friends. it is more clear now.

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It doesn't seem that there is any relation to the compactness of a manifold and its tangent bundle (unless its tangent bundle is compact iff its tangent bundle is non-trivial. This I highly doubt to be true).[edit- the tangent bundle of the moebius band would be the moebius band again wouldn't it? So this would be a counter-example to this obviously wrong claim].

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lavinia

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It doesn't seem that there is any relation to the compactness of a manifold and its tangent bundle (unless its tangent bundle is compact iff its tangent bundle is non-trivial. This I highly doubt to be true).[edit- the tangent bundle of the moebius band would be the moebius band again wouldn't it? So this would be a counter-example to this obviously wrong claim].

The tangent bundle is never compact. The tangent sphere bundle of a compact manifold is always compact.

The tangent bundle of the 2 sphere is not RP^3. The tangent circle bundle is RP^3.

The tangent bundle to the Moebius band is not the Moebius band. It is a 4 dimensional manifold. The Moebius band is a surface.

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Of course, you are right. I think that I read somewhere that the tangent sphere bundle of a sphere is RP^3 and my mind left out the "sphere" part (I must have somehow convinced myself that the bundle twists in such a way that it becomes non-compact :S). And the tangent bundle of a manifold of dimension n is a 2n dimensional manifold, so God knows where the Moebius band thing came from- sorry about that! :D

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quasar987

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a) closed

b) homeomorphic to R^n

Assume TM were compact and derive a contradiction.

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Ok, yeah, that's pretty obvious now :/ Thanks!

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