Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / B / 5 / Solution

One definition of $T_{x}M$ 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 $u^1,\dots ,u^n$, each has the unique form $v={\sum }_i{v}^i\partial /\partial u^i{|}_x$, identifying $T_{x}M$ 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 $\varphi :U\to {\mathbb{R}}^n$, give $TM{|}_U$ coordinates $(x,v)\mapsto (\varphi (x),D{\varphi }_{x}v)$. If $F=\psi \circ {\varphi }^{-1}$ is a base transition, the induced transition isThis 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 $n\ge 1$, 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 $v^j\ne 0$, use the ratios $v^i/v^j$ for $i\ne j$. Under a base change, the fibre line maps by $[v]\mapsto [D{F}_{u}v]$; 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 $S{O}_3$ 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 the stabilizer has order two, and $P(T{S}^2)\cong S{O}_3/C_2$.

This is an example of the projectivized tangent bundle.