TITLE: How do I see that the tangent bundle of torus is trivial
QUESTION [11 upvotes]: I've been having a hard time trying to determine if the tangent bundle of a differentiable manifold is trivial. Namely, if there exists a diffeomorphism between the tangent bundle $TM$ of a given manifold $M$ and the product manifold of $M\times \mathbb{R}^n$. 
I've managed to build a diffeomorphism from $TS^1$ to $S^1\times \mathbb{R}^1$. But the case with torus $S^1\times S^1$ seems harder, since the dimension is higher. 
In general, how do I show that $S^1\times \cdots \times S^1$ has trivial tangent bundle?

REPLY [16 votes]: Here are three ways.

Take one vector field in the direction of each factor to obtain a trivialization.  Work by induction.
$\mathbb{S}^1\times\cdots\times\mathbb{S}^1$ is a Lie group.  Prove that any Lie group is parallelizable.  (Take a left-invariant basis of the tangent space at $(1,1)$ and move it around by the Lie group's self-action.)
Prove that the product of two parallelizable manifolds is again parallelizable (see 1).  Corollary: Any product of circles is parallelizable.