= Solution
One definition of $T_xM$ is the space of derivations at $x$: linear maps $v:C^\infty(M)\to\mathbb R$ satisfying $v(fh)=f(x)v(h)+h(x)v(f)$. In local coordinates $u^1,\ldots,u^n$, each has the unique form $v=\sum_iv^i\partial/\partial u^i|_x$, identifying $T_xM$ with $\mathbb R^n$. Equivalently it is the space of velocities of curves through $x$, identifying curves with the same coordinate derivative.
For a chart $\phi:U\to\mathbb R^n$, give $TM|_U$ coordinates $(x,v)\mapsto(\phi(x),D\phi_xv)$. If $F=\psi\circ\phi^{-1}$ is a base transition, the induced transition is
$$
(u,v)\longmapsto(F(u),DF_u v).
$$
This is a smooth bijection with smooth inverse coming from the reverse chart. The resulting bundle atlas makes $TM$ a <smooth manifold> of dimension $2n$. These local product charts also give the usual <Hausdorff> and second-countable <topology>, inherited from a countable base atlas and its fibre coordinates.
For $n\ge1$, identify nonzero vectors in each fibre up to nonzero real scalar multiplication. Locally this gives $P(TM)|_U\cong U\times\mathbb{RP}^{n-1}$. In a projective chart where $v^j\ne0$, use the ratios $v^i/v^j$ for $i\ne j$. Under a base change, the fibre line maps by $[v]\mapsto[DF_uv]$; its projective coordinates are ratios of smooth linear expressions with nonzero denominator on the overlap. This defines a <smooth manifold> of dimension $n+(n-1)=2n-1$.
For the sphere, a point is a pair $(x,L)$ with $|x|=1$ and an unoriented line $L\subset x^\perp$. Choose a unit $v\in L$. The triple $(x,v,x\times v)$ is a positively oriented orthonormal frame. Given a second pair, the unique <orientation>-preserving linear map between chosen frames lies in $SO_3$ and carries the first pair to the second. Thus the action is transitive.
A rotation stabilizing $(x,L)$ fixes $x$, so rotates its tangent plane. To preserve the unoriented line it must act by $v\mapsto v$ or $v\mapsto-v$, giving exactly the identity and a half-turn about $x$. Hence \b[the <stabilizer> has order two], and $P(TS^2)\cong SO_3/C_2$.
This is an example of the <projectivized tangent bundle>.
Back to article page
