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I have a question about the tautological line bundle over $\mathbb R\mathbb P^n$. Recall, this bundle is that whose total space is $$E(\gamma_n):=\{([x], v)\in\mathbb R\mathbb P^n\times \mathbb R^{n+1}: v\in [x]\}.$$ The projection is the usual one.. How can I show the sections on this bundle are of the form $[x]\mapsto ([x], f(x)x)$ where $f:\mathbb S^n\rightarrow \mathbb R$ is continuous ad satisfies $f(-x)=-f(x)$? Any help will be valuable.. Thanks

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You have to add the condition $f(-x)=-f(x)$ –  Georges Elencwajg Sep 29 '13 at 10:57
    
yes you're right @GeorgesElencwajg, with this I'll prove this vector bundle is not trivial.. –  PtF Sep 29 '13 at 12:32

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