# What are examples of parallelizable manifolds, and why does parallelizable correspond to $TM$ being trivial?

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in M$.

1. What are few examples of parallelizable manifolds of dimension at least ? What is an example which is compact? What are some nonparalleizable ones?
2. How do I show that a manifold $M$ as above is parallelizable if and only if the tangent bundle $TM$ is trivial?

I am a total beginner with these smooth manifold things, so any help or references to a book that can help answer my quetsions would be greatly appreciated!

Some thoughts so far. I give some examples in $\mathbb{R}^3$. A cylinder is certainly paralleizable by taking vector fields $X_1$ constant, parallel to the direction of the axis and $X_2$ constant, pointing around the circumference of the cylinder. Another example is a torus; take vector fields $X_1$ constant, pointing along the circumference of the "big circle" and $X_2$ constant, looping in and out of the "donut hole." The torus is compact, and the cylinder is also compact if it is finite. As for nonparallelizable manifolds $S^2$ is an example; any attempt to parallelize $S^2$ leads to points on the sphere at which the vector field cannot be nonzero, which isn't good if we want to get the tangent spce at those points by combining tangent vectors from the vector fields. This is true by the Hairy Ball Theorem.

• The standard examples that are in each textbook include spheres: they are parallelizable in dimensions $0,1,3,7$ and not in other dimensions. The fact that these spheres are paralelizable can be shown by complex, quaternionic resp. octonionic multiplication of some fixed bases of $T_e S^k$ (realized in $\mathbb{R}^{k+1}$). In even-dimensional spheres, there is not even one nowhere zero vector field on the sphere ("Hairy ball theorem"). Dec 16 '14 at 22:44
• note that the examples you give (torus, cylinder) are lie groups, which are always parallelizable
– yoyo
Dec 16 '14 at 22:49

Some good $($introductory$)$ sources, in general, for all things smooth manifolds:

• Topology from the Differentiable Viewpoint, by Milnor
• Differential Topology, by Guillemin-Pollack
• Differential Forms and Applications, by Do Carmo
• A Comprehensive Introduction to Differential Geometry, Vol. 1, by Spivak
• Introduction to Smooth Manifolds, by Lee
• Foundations of Differentiable Manifolds and Lie Groups, by Warner
• Brian Conrad's Differential Geometry Notes