Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 26I (Differential Geometry)

Let $d=\dim X$ and choose a local parametrizationwhose derivative has rank $d$. DefineThis is a vector subspace because it is the image of a linear map, and rank $D{\varphi }_{u_0}=d$ gives $\dim T_{p}X=d$.

If $\psi$ is another local parametrization through $p$, the transition map $h={\psi }^{-1}\circ \varphi$ is a local diffeomorphism. The chain rule givesand $Dh$ is invertible, so $\mathrm{im}D\varphi =\mathrm{im}D\psi$. Thus the definition is independent of the parametrization. Equivalently, $T_{p}X$ is the set of velocity vectors of smooth curves in $X$ through $p$. This is the tangent space from a local parametrization.

After a fixed orthogonal change of ambient coordinates, a neighbourhood of $p$ in $X$ is the graphof a smooth map $g:U\to {\mathbb{R}}^{n-d}$. Write $p_i=(u_i,g(u_i))$ for all sufficiently large $i$. Every tangent vector has the unique formThe convergence of $w_i$ implies convergence of its first component, say $v_i\to v$. Since $u_i\to u$ and $Dg$ is continuous,This proves the closedness of tangent spaces under convergent base points.

Conversely, suppose $d>0$ and write $w=(v,Dg(u)v)\in T_{p}X$. Choose a nonzero $e\in {\mathbb{R}}^d$ and positive $t_i\to 0$ with $u_i=u+t_{i}e\in U$. Thenand $p_i\to p$, $w_i\to w$. Every tangent vector arises as such a limit from distinct base points.

If $a\ne 0$, the first equation defines the ellipsein the $(x_1,x_2)$-plane, a smooth one-dimensional manifold, while $x_3=a{x}_4$ defines a one-dimensional linear subspace in the $(x_3,x_4)$-plane. Their product is a smooth manifold, parametrized byso $\dim X_a=2$.

For $a=0$, positivity of the first equation forces $x_1=x_2=0$, and the second forces $x_3=0$. Hencea smooth line. Therefore

The answer is no. Let $a_i\ne 0$ tend to zero and takeAt a point $x\in X_a$ with $a\ne 0$, differentiating the defining equations givesThusfor every $i$, and $w_i\to w=(0,1,0,0)$. But $X_0$ is the $x_4$-axis, soHenceThe obstruction is the dimension jump in a smooth family of manifolds.