27
$\begingroup$

I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that $$\dim(TM)=2\dim(M).$$

Do the tangent bundles of the following spaces have any "known form", i.e. can be constructed (up to diffeomorphism) from known spaces $\mathbb{R}^n$, $\mathbb{S}^n$, $\mathbb{P}^n$, $\mathbb{T}^n$ via operations $\times$, $\#$, $\coprod$?

  • $T(\mathbb{S}^2)=?$
  • $T(\mathbb{T}^2)=?$
  • $T(\mathbb{T}^2\#T^2)=?$
  • $T(k\mathbb{T}^2)=?$, $\;\;\;k\in\mathbb{N}$ ($k$-fold connected sum $\#$)
  • $T(\mathbb{P}^2)=?$
  • $T(\mathbb{P}^2\#\mathbb{P}^2)=?$
  • $T(k\mathbb{P}^2)=?$, $\;\;\;k\in\mathbb{N}$ ($k$-fold connected sum $\#$)
  • $T(\mathbb{S}^n)=?$
  • $T(\mathbb{T}^n)=?$
  • $T(\mathbb{P}^n)=?$

($\mathbb{S}^n$ ... n-sphere, $\mathbb{T}^n$ ... $n$-torus $\mathbb{S}^1\times\ldots\times\mathbb{S}^1$, $\mathbb{P}^n$ ... real projective $n$-space, $\#$ ... connected sum)

I'm making these examples up, so if there are more illustrative ones, please explain those.

BTW, I know that $T(\mathbb{S}^1)=\mathbb{S}^1\times\mathbb{R}$ by visually thinking about it.

P.S. I'm just learning about these notions...

ADDITION: I just realized that all Lie groups have trivial tangent bundle, so $T(\mathbb{T}^n)\approx\mathbb{T}^n\!\times\!\mathbb{R}^n$.

  • 1
    Why did you add the Riemannian and symplectic tags? You didn't specify Riemannian structures and the tangent bundle is not naturally a symplectic manifold (as opposed to the cotangent bundle).2011-06-19
  • 0
    Isn't riemannian geometry an upgrade (additional structure) of differential topology? And isn't Symplectic Geometry an upgrade of Riemannian? If those tags are inappropriate, I can remove them...2011-06-19
  • 3
    I would say Riemannian geometry is a subdiscipline of differential geometry (not topology). Every (paracompact) manifold can be equipped with a Riemannian metric and indeed, this is additional structure. On the other hand, symplectic geometry is a different business: only even-dimensional manifolds admit a symplectic structure and among the spheres only $S^2$ does. As I said, the cotangent bundle of a manifold has a canonical symplectic structure. I think removing those tags would be better.2011-06-19
  • 0
    Also: what is $g$ in $T(g\mathbb{T}^2)$ and $T(g\mathbb{P}^2)$? I suppose you mean the $g$-fold connected sum, right? By the way: in differential geometry $g$ usually stands for a Riemannian metric and in differential topology for the genus of a $2$-manifold, so I would choose a different letter.2011-06-19
  • 0
    Yes, $g\mathbb{P}^2=\mathbb{P}^2\#\ldots\#\mathbb{P}^2$ $g$-times. I included the tags because smooth manifolds are the central object of all those fields. OK, I'll remove them.2011-06-19

3 Answers 3