Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 25I (Differential Geometry)

Suppose $S$ lies in the closed ball of radius $R$ centred at $c$. By compactness, the continuous function $q\mapsto |q-c|$ attains its maximum $r\le R$ at some $p\in S$. The sphere of radius $r$ about $c$ is a supporting sphere tangent to $S$ at $p$.

Put $N=(p-c)/r$. For any unit vector $v\in T_{p}S$, choose a surface curve $\alpha$ with $\alpha (0)=p$ and ${\alpha }^{\prime }(0)=v$. Since $|\alpha (t)-c{|}^2$ has a local maximum at zero,The scalar $⟨{\alpha }^{\prime \prime }(0),N⟩$ is the normal curvature in direction $v$, so every normal curvature is at most $-1/r$ with this choice of unit normal. Applying this to principal directions shows that both principal curvatures satisfy $k_i\le -1/r$. ThereforeThis proves both assertions, including the existence of an elliptic point on every compact regular surface.

Choose an orthonormal basis $(e_1,e_2)$ of $T_{p}S$ and write $e(\theta )=\cos \theta e_1+\sin \theta e_2$. On a punctured disc on which the exponential map is a diffeomorphism, the mapdefines geodesic polar coordinates centred at $p$. The curves $r\mapsto X(r,\theta )$ are unit-speed geodesics, so the first coefficient of the first fundamental form isThe Gauss lemma says that radial and angular coordinate vectors are orthogonal, hence

The angular variation $J=X_{\theta }$ is a Jacobi field along each radial geodesic, with initial dataSince $G=⟨X_{\theta },X_{\theta }⟩=|J{|}^2$, smooth dependence of the geodesic flow givesConsequentlyas required.

Put $h(r,\theta )=\sqrt{G(r,\theta )}$. The Jacobi equation in geodesic polar coordinates and the initial conditions from part (b) areInside a geodesic polar coordinate ball, $h>0$. If $K\le 0$, then $h_{rr}\ge 0$, so $h_r\ge 1$ and therefore $h(r,\theta )\ge r$. The Riemannian area element is $hdrd\theta$, whenceThus nonpositive Gaussian curvature makes such a ball at least as large as the Euclidean disk of the same radius.

Again set $h=\sqrt{G}$ and fix $\theta$. For $0<\alpha <C^{-1/2}$ defineSince $h^{\prime \prime }=-Kh$ and $K\le C$,For $0<r<\pi /\sqrt{C}$, differentiation givesThe expression in parentheses vanishes at zero, and henceIts limit at zero is $(1-\alpha \sqrt{C})/\sqrt{C}>0$, so $f>0$. Letting $\alpha \uparrow C^{-1/2}$ yields the one-dimensional Rauch comparison inequalityIt follows thatNowas $x\to 0$. Given $0<\gamma <1$, choose $x_0\in (0,\pi )$ such that $2(1-\cos x)/x^2\ge \gamma$ for $0<x\le x_0$, and set ${\epsilon }_0=x_0/\sqrt{C}$. Then every geodesic polar coordinate ball of radius $\epsilon \le {\epsilon }_0$ satisfies