Tangent spaces at different points on a manifold

[WWGD]

The tangent bundle is never a vector space because fibers at different points can not be added. In the case of trivial bundles, adding vectors from different fibers forgets the bundle structure. It is no longer a bundle.

If the manifold is not homeomorphic to $R^n$ then it cannot be a vector space, even if its tangent bundle is trivial For instance, the tangent bundle of every orientable compact 3 manifold is trivial. But no compact space can be a vector space.

I understand, but the OP wanted to know how to add tangent vectors at different points, not points themselves. Tangent vectors live in the tangent bundle. And Isn't the tangent bundle of $\mathbb R^n$ equal to $\mathbb R^{2n}$?

 

[lavinia]

And Isn't the tangent bundle of $\mathbb R^n$ equal to $\mathbb R^{2n}$?

It is homeomorphic to $R^{2n}$ but not isomorphic as a bundle. By itself $R^{2n}$ is not a bundle.

 

[WWGD]

It is homeomorphic to $R^{2n}$ but not isomorphic as a bundle. By itself $R^{2n}$ is not a bundle.

And that was the argument I was presenting for why one cannot add tangent vectors at different points: the tangent bundle is not a vector space.

 

[lavinia]

Maybe a new thread should be started that discusses the theory of vector bundles - with structure group the general linear group - for manifolds. this is a highly researched area with many difficult Theorems but we could

And that was the argument I was presenting for why one cannot add tangent vectors at different points: the tangent bundle is not a vector space.

yes.