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Let $M$ a (real) $n$-dimensional connected differentiable manifold.

(a) The tangent bundle $TM$ is trivial, $TM \simeq M \times \mathbb R^n$;

(b) $M$ is orientable.

Consider the statements (a) $\Rightarrow$ (b), (b) $\Rightarrow$ (a). Which of them are true?

Well, I'm quite sure (a) $\Rightarrow$ (b) is true. What about (b) $\Rightarrow$ (a)? I can't believe it is true... in that case we would have (a) $\Leftrightarrow$ (b) which seems very strange to me.

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up vote 5 down vote accepted

The sphere $\rm S^2$ is an orientable manifold, but $\rm TS^2$ is not trivial !

But the contrary is true, a manifold with trivial tangent bundle is orientable.

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I feel so stupid. It was extremely easy, thanks. – Romeo Feb 15 '13 at 20:01

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