= Real tautological line bundle
{title2=$\gamma_n$}
= Canonical real line bundle over real projective space
{synonym}
Over <Real projective space> $\mathbb{RP}^n$, the real tautological line bundle is
$$
\gamma_n=\{(\ell,v):v\in\ell\}\subseteq\mathbb{RP}^n\times\mathbb R^{n+1}.
$$
On the standard chart $U_i=\{[x]:x_i\ne0\}$, let $s_i([x])=x/x_i$. A <local trivialization> sends $([x],t)$ to $([x],t s_i([x]))$, with inverse $([x],v)\mapsto([x],v_i)$. These maps prove that this is a <real line bundle>. Its unit <sphere bundle> is the <antipodal map> quotient $S^n\to\mathbb{RP}^n$. Its <mod-two Euler class of a real line bundle> is the degree-one generator of the <mod-two cohomology ring of real projective space>.
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