= Projectivized tangent bundle
{title2=$P(TM)$}
The projectivized <tangent bundle> of a positive-dimensional smooth <manifold> is the bundle of unoriented one-dimensional subspaces of its <tangent spaces>. A local tangent trivialization gives $P(TM)|_U\cong U\times\mathbb{RP}^{n-1}$. If a base-chart transition is $F$, the line coordinates transform by $[v]\mapsto[DF_xv]$. In any projective chart, this is a ratio of smooth linear expressions with nonzero denominator. It therefore defines a smooth manifold of dimension $2n-1$.
For the two-sphere, a point consists of $x\in S^2$ and a line $L\subset x^\perp$. Choosing a unit vector $v\in L$ gives an oriented <orthonormal basis> $(x,v,x\times v)$. A unique <rotation> takes one chosen frame to another, so $SO_3$ acts transitively. Its <stabilizer> consists of the identity and the half-turn about $x$, which reverses $v$ while preserving its unoriented line. Thus $P(TS^2)\cong SO_3/C_2$.
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