# Differential Geometry

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Paper 1, Section II, 26F

comment(a) Let $S \subset \mathbb{R}^{3}$ be a surface. Give a parametrisation-free definition of the first fundamental form of $S$. Use this definition to derive a description of it in terms of the partial derivatives of a local parametrisation $\phi: U \subset \mathbb{R}^{2} \rightarrow S$.

(b) Let $a$ be a positive constant. Show that the half-cone

$\Sigma=\left\{(x, y, z) \mid z^{2}=a\left(x^{2}+y^{2}\right), z>0\right\}$

is locally isometric to the Euclidean plane. [Hint: Use polar coordinates on the plane.]

(c) Define the second fundamental form and the Gaussian curvature of $S$. State Gauss' Theorema Egregium. Consider the set

$V=\left\{(x, y, z) \mid x^{2}+y^{2}+z^{2}-2 x y-2 y z=0\right\} \backslash\{(0,0,0)\} \subset \mathbb{R}^{3}$

(i) Show that $V$ is a surface.

(ii) Calculate the Gaussian curvature of $V$ at each point. [Hint: Complete the square.]

Paper 2, Section II, $26 \mathbf{F}$

commentLet $U$ be a domain in $\mathbb{R}^{2}$, and let $\phi: U \rightarrow \mathbb{R}^{3}$ be a smooth map. Define what it means for $\phi$ to be an immersion. What does it mean for an immersion to be isothermal?

Write down a formula for the mean curvature of an immersion in terms of the first and second fundamental forms. What does it mean for an immersed surface to be minimal? Assume that $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ is an isothermal immersion. Prove that it is minimal if and only if $x, y, z$ are harmonic functions of $u, v$.

For $u \in \mathbb{R}, v \in[0,2 \pi]$, and smooth functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$, assume that

$\phi(u, v)=(f(u) \cos v, f(u) \sin v, g(u))$

is an isothermal immersion. Find all possible pairs $(f, g)$ such that this immersion is minimal.

Paper 3, Section II, F

commentLet $X$ and $Y$ be smooth boundaryless manifolds. Suppose $f: X \rightarrow Y$ is a smooth map. What does it mean for $y \in Y$ to be a regular value of $f$ ? State Sard's theorem and the stack-of-records theorem.

Suppose $g: X \rightarrow Y$ is another smooth map. What does it mean for $f$ and $g$ to be smoothly homotopic? Assume now that $X$ is compact, and has the same dimension as $Y$. Suppose that $y \in Y$ is a regular value for both $X$ and $Y$. Prove that

$\# f^{-1}(y)=\# g^{-1}(y)(\bmod 2)$

Let $U \subset S^{n}$ be a non-empty open subset of the sphere. Suppose that $h: S^{n} \rightarrow S^{n}$ is a smooth map such that $\# h^{-1}(y)=1(\bmod 2)$ for all $y \in U$. Show that there must exist a pair of antipodal points on $S^{n}$ which is mapped to another pair of antipodal points by $h$.

[You may assume results about compact 1-manifolds provided they are accurately stated.]

Paper 4, Section II, F

commentLet $I \subset \mathbb{R}$ be an interval, and $S \subset \mathbb{R}^{3}$ be a surface. Assume that $\alpha: I \rightarrow S$ is a regular curve parametrised by arc-length. Define the geodesic curvature of $\alpha$. What does it mean for $\alpha$ to be a geodesic curve?

State the global Gauss-Bonnet theorem including boundary terms.

Suppose that $S \subset \mathbb{R}^{3}$ is a surface diffeomorphic to a cylinder. How large can the number of simple closed geodesics on $S$ be in each of the following cases?

(i) $S$ has Gaussian curvature everywhere zero;

(ii) $S$ has Gaussian curvature everywhere positive;

(iii) $S$ has Gaussian curvature everywhere negative.

In cases where there can be two or more simple closed geodesics, must they always be disjoint? Justify your answer.

[A formula for the Gaussian curvature of a surface of revolution may be used without proof if clearly stated. You may also use the fact that a piecewise smooth curve on a cylinder without self-intersections either bounds a domain homeomorphic to a disc or is homotopic to the waist-curve of the cylinder.]

Paper 1, Section II, I

comment(a) Let $X \subset \mathbb{R}^{N}$ be a manifold. Give the definition of the tangent space $T_{p} X$ of $X$ at a point $p \in X$.

(b) Show that $X:=\left\{-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-1\right\} \cap\left\{x_{0}>0\right\}$ defines a submanifold of $\mathbb{R}^{4}$ and identify explicitly its tangent space $T_{\mathbf{x}} X$ for any $\mathbf{x} \in X$.

(c) Consider the matrix group $O(1,3) \subset \mathbb{R}^{4^{2}}$ consisting of all $4 \times 4$ matrices $A$ satisfying

$A^{t} M A=M$

where $M$ is the diagonal $4 \times 4$ matrix $M=\operatorname{diag}(-1,1,1,1)$.

(i) Show that $O(1,3)$ forms a group under matrix multiplication, i.e. it is closed under multiplication and every element in $O(1,3)$ has an inverse in $O(1,3)$.

(ii) Show that $O(1,3)$ defines a 6-dimensional manifold. Identify the tangent space $T_{A} O(1,3)$ for any $A \in O(1,3)$ as a set $\{A Y\}_{Y \in \mathfrak{S}}$ where $Y$ ranges over a linear subspace $\mathfrak{S} \subset \mathbb{R}^{4^{2}}$ which you should identify explicitly.

(iii) Let $X$ be as defined in (b) above. Show that $O^{+}(1,3) \subset O(1,3)$ defined as the set of all $A \in O(1,3)$ such that $A \mathbf{x} \in X$ for all $\mathbf{x} \in X$ is both a subgroup and a submanifold of full dimension.

[You may use without proof standard theorems from the course concerning regular values and transversality.]

Paper 2, Section II, I

comment(a) State the fundamental theorem for regular curves in $\mathbb{R}^{3}$.

(b) Let $\alpha: \mathbb{R} \rightarrow \mathbb{R}^{3}$ be a regular curve, parameterised by arc length, such that its image $\alpha(\mathbb{R}) \subset \mathbb{R}^{3}$ is a one-dimensional submanifold. Suppose that the set $\alpha(\mathbb{R})$ is preserved by a nontrivial proper Euclidean motion $\phi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$.

Show that there exists $\sigma_{0} \in \mathbb{R}$ corresponding to $\phi$ such that $\phi(\alpha(s))=\alpha\left(\pm s+\sigma_{0}\right)$ for all $s \in \mathbb{R}$, where the choice of $\pm \operatorname{sign}$ is independent of $s$. Show also that the curvature $k(s)$ and torsion $\tau(s)$ of $\alpha$ satisfy

$\begin{gathered} k\left(\pm s+\sigma_{0}\right)=k(s) \text { and } \\ \tau\left(\pm s+\sigma_{0}\right)=\tau(s) \end{gathered}$

with equation (2) valid only for $s$ such that $k(s)>0$. In the case where the sign is $+$ and $\sigma_{0}=0$, show that $\alpha(\mathbb{R})$ is a straight line.

(c) Give an explicit example of a curve $\alpha$ satisfying the requirements of (b) such that neither of $k(s)$ and $\tau(s)$ is a constant function, and such that the curve $\alpha$ is closed, i.e. such that $\alpha(s)=\alpha\left(s+s_{0}\right)$ for some $s_{0}>0$ and all $s$. [Here a drawing would suffice.]

(d) Suppose now that $\alpha: \mathbb{R} \rightarrow \mathbb{R}^{3}$ is an embedded regular curve parameterised by arc length $s$. Suppose further that $k(s)>0$ for all $s$ and that $k(s)$ and $\tau(s)$ satisfy (1) and (2) for some $\sigma_{0}$, where the choice $\pm$ is independent of $s$, and where $\sigma_{0} \neq 0$ in the case of + sign. Show that there exists a nontrivial proper Euclidean motion $\phi$ such that the set $\alpha(\mathbb{R})$ is preserved by $\phi$. [You may use the theorem of part (a) without proof.]

Paper 3, Section II, I

comment(a) Show that for a compact regular surface $S \subset \mathbb{R}^{3}$, there exists a point $p \in S$ such that $K(p)>0$, where $K$ denotes the Gaussian curvature. Show that if $S$ is contained in a closed ball of radius $R$ in $\mathbb{R}^{3}$, then there is a point $p$ such that $K(p) \geqslant R^{-2}$.

(b) For a regular surface $S \subset \mathbb{R}^{3}$, give the definition of a geodesic polar coordinate system at a point $p \in S$. Show that in such a coordinate system, $\lim _{r \rightarrow 0} G(r, \theta)=0$, $\lim _{r \rightarrow 0}(\sqrt{G})_{r}(r, \theta)=1, E(r, \theta)=1$ and $F(r, \theta)=0$. [You may use without proof standard properties of the exponential map provided you state them clearly.]

(c) Let $S \subset \mathbb{R}^{3}$ be a regular surface. Show that if $K \leqslant 0$, then any geodesic polar coordinate ball $B\left(p, \epsilon_{0}\right) \subset S$ of radius $\epsilon_{0}$ around $p$ has area satisfying

$\text { Area } B\left(p, \epsilon_{0}\right) \geqslant \pi \epsilon_{0}^{2}$

[You may use without proof the identity $(\sqrt{G})_{r r}(r, \theta)=-\sqrt{G} K$.]

(d) Let $S \subset \mathbb{R}^{3}$ be a regular surface, and now suppose $-\infty<K \leqslant C$ for some constant $0<C<\infty$. Given any constant $0<\gamma<1$, show that there exists $\epsilon_{0}>0$, depending only on $C$ and $\gamma$, so that if $B(p, \epsilon) \subset S$ is any geodesic polar coordinate ball of radius $\epsilon \leqslant \epsilon_{0}$, then

$\text { Area } B(p, \epsilon) \geqslant \gamma \pi \epsilon^{2}$

[Hint: For any fixed $\theta_{0}$, consider the function $f(r):=\sqrt{G}\left(r, \theta_{0}\right)-\alpha \sin (\sqrt{C} r)$, for all $0<$ $\alpha<\frac{1}{\sqrt{C}}$. Derive the relation $f^{\prime \prime} \geqslant-C f$ and show $f(r)>0$ for an appropriate range of $r .$ The following variant of Wirtinger's inequality may be useful and can be assumed without proof: if $g$ is a $C^{1}$ function on $[0, L]$ vanishing at 0 , then $\int_{0}^{L}|g(x)|^{2} d x \leqslant \frac{L}{2 \pi} \int_{0}^{L}\left|g^{\prime}(x)\right|^{2} d x$.]

Paper 4, Section II, I

comment(a) State the Gauss-Bonnet theorem for compact regular surfaces $S \subset \mathbb{R}^{3}$ without boundary. Identify all expressions occurring in any formulae.

(b) Let $S \subset \mathbb{R}^{3}$ be a compact regular surface without boundary and suppose that its Gaussian curvature $K(x) \geqslant 0$ for all $x \in S$. Show that $S$ is diffeomorphic to the sphere.

Let $S_{n}$ be a sequence of compact regular surfaces in $\mathbb{R}^{3}$ and let $K_{n}(x)$ denote the Gaussian curvature of $S_{n}$ at $x \in S_{n}$. Suppose that

$\limsup _{n \rightarrow \infty} \inf _{x \in S_{n}} K_{n}(x) \geqslant 0$

(c) Give an example to show that it does not follow that for all sufficiently large $n$ the surface $S_{n}$ is diffeomorphic to the sphere.

(d) Now assume, in addition to $(\star)$, that all of the following conditions hold:

(1) There exists a constant $R<\infty$ such that for all $n, S_{n}$ is contained in a ball of radius $R$ around the origin.

(2) There exists a constant $M<\infty$ such that $\operatorname{Area}\left(S_{n}\right) \leqslant M$ for all $n$.

(3) There exists a constant $\epsilon_{0}>0$ such that for all $n$, all points $p \in S_{n}$ admit a geodesic polar coordinate system centred at $p$ of radius at least $\epsilon_{0}$.

(4) There exists a constant $C<\infty$ such that on all such geodesic polar neighbourhoods, $\left|\partial_{r} K_{n}\right| \leqslant C$ for all $n$, where $r$ denotes a geodesic polar coordinate.

(i) Show that for all sufficiently large $n$, the surface $S_{n}$ is diffeomorphic to the sphere. [Hint: It may be useful to identify a geodesic polar ball $B\left(p_{n}, \epsilon_{0}\right)$ in each $S_{n}$ for which $\int_{B\left(p_{n}, \epsilon_{0}\right)} K_{n} d A$ is bounded below by a positive constant independent of $n$.]

(ii) Explain how your example from (c) fails to satisfy one or more of these extra conditions (1)-(4).

[You may use without proof the standard computations for geodesic polar coordinates: $E=1, F=0, \lim _{r \rightarrow 0} G(r, \theta)=0, \lim _{r \rightarrow 0}(\sqrt{G})_{r}(r, \theta)=1$, and $\left.(\sqrt{G})_{r r}=-K \sqrt{G} .\right]$

Paper 1, Section II, H

commentLet $n \geqslant 1$ be an integer.

(a) Show that $\mathbb{S}^{n}=\left\{x \in \mathbb{R}^{n+1}: x_{1}^{2}+\cdots+x_{n+1}^{2}=1\right\}$ defines a submanifold of $\mathbb{R}^{n+1}$ and identify explicitly its tangent space $T_{x} \mathbb{S}^{n}$ for any $x \in \mathbb{S}^{n}$.

(b) Show that the matrix group $S O(n) \subset \mathbb{R}^{n^{2}}$ defines a submanifold. Identify explicitly the tangent space $T_{R} S O(n)$ for any $R \in S O(n)$.

(c) Given $v \in \mathbb{S}^{n}$, show that the set $S_{v}=\{R \in S O(n+1): R v=v\}$ defines a submanifold $S_{v} \subset S O(n+1)$ and compute its dimension. For $v \neq w$, is it ever the case that $S_{v}$ and $S_{w}$ are transversal?

[You may use standard theorems from the course concerning regular values and transversality.]

Paper 2, Section II, H

comment(a) Let $\alpha:(a, b) \rightarrow \mathbb{R}^{3}$ be a smooth regular curve parametrised by arclength. For $s \in(a, b)$, define the curvature $k(s)$ and (where defined) the torsion $\tau(s)$ of $\alpha$. What condition must be satisfied in order for the torsion to be defined? Derive the Frenet equations.

(b) If $\tau(s)$ is defined and equal to 0 for all $s \in(a, b)$, show that $\alpha$ lies in a plane.

(c) State the fundamental theorem for regular curves in $\mathbb{R}^{3}$, giving necessary and sufficient conditions for when curves $\alpha(s)$ and $\widetilde{\alpha}(s)$ are related by a proper Euclidean motion.

(d) Now suppose that $\widetilde{\alpha}:(a, b) \rightarrow \mathbb{R}^{3}$ is another smooth regular curve parametrised by arclength, and that $\widetilde{k}(s)$ and $\tilde{\tau}(s)$ are its curvature and torsion. Determine whether the following statements are true or false. Justify your answer in each case.

(i) If $\tau(s)=0$ whenever it is defined, then $\alpha$ lies in a plane.

(ii) If $\tau(s)$ is defined and equal to 0 for all but one value of $s$ in $(a, b)$, then $\alpha$ lies in a plane.

(iii) If $k(s)=\tilde{k}(s)$ for all $s, \tau(s)$ and $\tilde{\tau}(s)$ are defined for all $s \neq s_{0}$, and $\tau(s)=\tilde{\tau}(s)$ for all $s \neq s_{0}$, then $\alpha$ and $\widetilde{\alpha}$are related by a rigid motion.

Paper 3, Section II, H

comment(a) Let $\alpha:(a, b) \rightarrow \mathbb{R}^{2}$ be a regular curve without self intersection given by $\alpha(v)=(f(v), g(v))$ with $f(v)>0$ for $v \in(a, b)$.

Consider the local parametrisation given by

$\phi:(0,2 \pi) \times(a, b) \rightarrow \mathbb{R}^{3}$

where $\phi(u, v)=(f(v) \cos u, f(v) \sin u, g(v))$.

(i) Show that the image $\phi((0,2 \pi) \times(a, b))$ defines a regular surface $S$ in $\mathbb{R}^{3}$.

(ii) If $\gamma(s)=\phi(u(s), v(s))$ is a geodesic in $S$ parametrised by arc length, then show that $f(v(s))^{2} u^{\prime}(s)$ is constant in $s$. If $\theta(s)$ denotes the angle that the geodesic makes with the parallel $S \cap\{z=g(v(s))\}$, then show that $f(v(s)) \cos \theta(s)$ is constant in $s$.

(b) Now assume that $\alpha(v)=(f(v), g(v))$ extends to a smooth curve $\alpha:[a, b] \rightarrow \mathbb{R}^{2}$ such that $f(a)=0, f(b)=0, f^{\prime}(a) \neq 0, f^{\prime}(b) \neq 0$. Let $\bar{S}$ be the closure of $S$ in $\mathbb{R}^{3}$.

(i) State a necessary and sufficient condition on $\alpha(v)$ for $\bar{S}$ to be a compact regular surface. Justify your answer.

(ii) If $\bar{S}$ is a compact regular surface, and $\gamma:(-\infty, \infty) \rightarrow \bar{S}$ is a geodesic, show that there exists a non-empty open subset $U \subset \bar{S}$ such that $\gamma((-\infty, \infty)) \cap U=\emptyset$.

Paper 4, Section II, H

comment(a) Let $\gamma:(a, b) \rightarrow \mathbb{R}^{2}$ be a regular curve without self-intersection given by $\gamma(v)=(f(v), g(v))$ with $f(v)>0$ for $v \in(a, b)$ and let $S$ be the surface of revolution defined globally by the parametrisation

$\phi:(0,2 \pi) \times(a, b) \rightarrow \mathbb{R}^{3}$

where $\phi(u, v)=(f(v) \cos u, f(v) \sin u, g(v))$, i.e. $S=\phi((0,2 \pi) \times(a, b))$. Compute its mean curvature $H$ and its Gaussian curvature $K$.

(b) Define what it means for a regular surface $S \subset \mathbb{R}^{3}$ to be minimal. Give an example of a minimal surface which is not locally isometric to a cone, cylinder or plane. Justify your answer.

(c) Let $S$ be a regular surface such that $K \equiv 1$. Is it necessarily the case that given any $p \in S$, there exists an open neighbourhood $\mathcal{U} \subset S$ of $p$ such that $\mathcal{U}$ lies on some sphere in $\mathbb{R}^{3}$ ? Justify your answer.

Paper 1, Section II, I

comment(a) Let $X \subset \mathbb{R}^{n}$ be a manifold and $p \in X$. Define the tangent space $T_{p} X$ and show that it is a vector subspace of $\mathbb{R}^{n}$, independent of local parametrization, of dimension equal to $\operatorname{dim} X$.

(b) Now show that $T_{p} X$ depends continuously on $p$ in the following sense: if $p_{i}$ is a sequence in $X$ such that $p_{i} \rightarrow p \in X$, and $w_{i} \in T_{p_{i}} X$ is a sequence such that $w_{i} \rightarrow w \in \mathbb{R}^{n}$, then $w \in T_{p} X$. If $\operatorname{dim} X>0$, show that all $w \in T_{p} X$ arise as such limits where $p_{i}$ is a sequence in $X \backslash p$.

(c) Consider the set $X_{a} \subset \mathbb{R}^{4}$ defined by $X_{a}=\left\{x_{1}^{2}+2 x_{2}^{2}=a^{2}\right\} \cap\left\{x_{3}=a x_{4}\right\}$, where $a \in \mathbb{R}$. Show that, for all $a \in \mathbb{R}$, the set $X_{a}$ is a smooth manifold. Compute its dimension.

(d) For $X_{a}$ as above, does $T_{p} X_{a}$ depend continuously on $p$ and $a$ for all $a \in \mathbb{R}$ ? In other words, let $a_{i} \in \mathbb{R}, p_{i} \in X_{a_{i}}$ be sequences with $a_{i} \rightarrow a \in \mathbb{R}, p_{i} \rightarrow p \in X_{a}$. Suppose that $w_{i} \in T_{p_{i}} X_{a_{i}}$ and $w_{i} \rightarrow w \in \mathbb{R}^{4}$. Is it necessarily the case that $w \in T_{p} X_{a}$ ? Justify your answer.

Paper 2, Section II, I

commentLet $\gamma(t):[a, b] \rightarrow \mathbb{R}^{3}$ denote a regular curve.

(a) Show that there exists a parametrization of $\gamma$ by arc length.

(b) Under the assumption that the curvature is non-zero, define the torsion of $\gamma$. Give an example of two curves $\gamma_{1}$ and $\gamma_{2}$ in $\mathbb{R}^{3}$ whose curvature (as a function of arc length $s$ ) coincides and is non-vanishing, but for which the curves are not related by a rigid motion, i.e. such that $\gamma_{1}(s)$ is not identically $\rho_{(R, T)}\left(\gamma_{2}(s)\right)$ where $R \in S O(3), T \in \mathbb{R}^{3}$ and

$\rho_{(R, T)}(v):=T+R v$

(c) Give an example of a simple closed curve $\gamma$, other than a circle, which is preserved by a non-trivial rigid motion, i.e. which satisfies

$\rho_{(R, T)}(v) \in \gamma([a, b]) \text { for all } v \in \gamma([a, b])$

for some choice of $R \in S O(3), T \in \mathbb{R}^{3}$ with $(R, T) \neq(\mathrm{Id}, 0)$. Justify your answer.

(d) Now show that a simple closed curve $\gamma$ which is preserved by a nontrivial smooth 1-parameter family of rigid motions is necessarily a circle, i.e. show the following:

Let $(R, T):(-\epsilon, \epsilon) \rightarrow S O(3) \times \mathbb{R}^{3}$ be a regular curve. If for all $\tilde{t} \in(-\epsilon, \epsilon)$,

$\rho_{(R(\tilde{t}), T(\tilde{t}))}(v) \in \gamma([a, b]) \text { for all } v \in \gamma([a, b]) \text {, }$

then $\gamma([a, b])$ is a circle. [You may use the fact that the set of fixed points of a non-trivial rigid motion is either $\emptyset$ or a line $L \subset \mathbb{R}^{3}$.]

Paper 3, Section II, I

commentLet $S \subset \mathbb{R}^{3}$ be a surface.

(a) Define the Gaussian curvature $K$ of $S$ in terms of the coefficients of the first and second fundamental forms, computed with respect to a local parametrization $\phi(u, v)$ of $S$.

Prove the Theorema Egregium, i.e. show that the Gaussian curvature can be expressed entirely in terms of the coefficients of the first fundamental form and their first and second derivatives with respect to $u$ and $v$.

(b) State the global Gauss-Bonnet theorem for a compact orientable surface $S$.

(c) Now assume that $S$ is non-compact and diffeomorphic to $\mathbb{S}^{2} \backslash\{(1,0,0)\}$ but that there is a point $p \in \mathbb{R}^{3}$ such that $S \cup\{p\}$ is a compact subset of $\mathbb{R}^{3}$. Is it necessarily the case that $\int_{S} K d A=4 / \pi ?$ Justify your answer.

Paper 4, Section II, I

commentLet $S \subset \mathbb{R}^{3}$ be a surface.

(a) Define what it means for a curve $\gamma: I \rightarrow S$ to be a geodesic, where $I=(a, b)$ and $-\infty \leqslant a<b \leqslant \infty$.

(b) A geodesic $\gamma: I \rightarrow S$ is said to be maximal if any geodesic $\widetilde{\gamma}: \tilde{I} \rightarrow S$ with $I \subset \tilde{I}$ and $\left.\tilde{\gamma}\right|_{I}=\gamma$ satisfies $I=\tilde{I}$. A surface is said to be geodesically complete if all maximal geodesics are defined on $I=(-\infty, \infty)$, otherwise, the surface is said to be geodesically incomplete. Give an example, with justification, of a non-compact geodesically complete surface $S$ which is not a plane.

(c) Assume that along any maximal geodesic

$\gamma:\left(-T_{-}, T_{+}\right) \rightarrow S$

the following holds:

$\tag{*} T_{\pm}<\infty \Longrightarrow \limsup _{s \rightarrow T_{\pm}}|K(\gamma(\pm s))|=\infty$

Here $K$ denotes the Gaussian curvature of $S$.

(i) Show that $S$ is inextendible, i.e. if $\widetilde{S} \subset \mathbb{R}^{3}$ is a connected surface with $S \subset \widetilde{S}$, then $\widetilde{S}=S$.

(ii) Give an example of a surface $S$ which is geodesically incomplete and satisfies $(*)$. Do all geodesically incomplete inextendible surfaces satisfy $(*)$ ? Justify your answer.

[You may use facts about geodesics from the course provided they are clearly stated.]

Paper 1, Section II, I

commentDefine what it means for a subset $X \subset \mathbb{R}^{N}$ to be a manifold.

For manifolds $X$ and $Y$, state what it means for a map $f: X \rightarrow Y$ to be smooth. For such a smooth map, and $x \in X$, define the differential map $d f_{x}$.

What does it mean for $y \in Y$ to be a regular value of $f$ ? Give an example of a map $f: X \rightarrow Y$ and a $y \in Y$ which is not a regular value of $f$.

Show that the set $S L_{n}(\mathbb{R})$ of $n \times n$ real-valued matrices with determinant 1 can naturally be viewed as a manifold $S L_{n}(\mathbb{R}) \subset \mathbb{R}^{n^{2}}$. What is its dimension? Show that matrix multiplication $f: S L_{n}(\mathbb{R}) \times S L_{n}(\mathbb{R}) \rightarrow S L_{n}(\mathbb{R})$, defined by $f(A, B)=A B$, is smooth. [Standard theorems may be used without proof if carefully stated.] Describe the tangent space of $S L_{n}(\mathbb{R})$ at the identity $I \in S L_{n}(\mathbb{R})$ as a subspace of $\mathbb{R}^{n^{2}}$.

Show that if $n \geqslant 2$ then the set of real-valued matrices with determinant 0 , viewed as a subset of $\mathbb{R}^{n^{2}}$, is not a manifold.

Paper 2, Section II, I

commentLet $\alpha: I \rightarrow \mathbb{R}^{3}$ be a regular smooth curve. Define the curvature $k$ and torsion $\tau$ of $\alpha$ and derive the Frenet formulae. Give the assumption which must hold for torsion to be well-defined, and state the Fundamental Theorem for curves in $\mathbb{R}^{3}$.

Let $\alpha$ be as above and $\tilde{\alpha}: I \rightarrow \mathbb{R}^{3}$ be another regular smooth curve with curvature $\tilde{k}$ and torsion $\tilde{\tau}$. Suppose $\tilde{k}(s)=k(s) \neq 0$ and $\tilde{\tau}(s)=\tau(s)$ for all $s \in I$, and that there exists a non-empty open subinterval $J \subset I$ such that $\left.\tilde{\alpha}\right|_{J}=\left.\alpha\right|_{J}$. Show that $\tilde{\alpha}=\alpha$.

Now let $S \subset \mathbb{R}^{3}$ be an oriented surface and let $\alpha: I \rightarrow S \subset \mathbb{R}^{3}$ be a regular smooth curve contained in $S$. Define normal curvature and geodesic curvature. When is $\alpha$ a geodesic? Give an example of a surface $S$ and a geodesic $\alpha$ whose normal curvature vanishes identically. Must such a surface $S$ contain a piece of a plane? Can such a geodesic be a simple closed curve? Justify your answers.

Show that if $\alpha$ is a geodesic and the Gaussian curvature of $S$ satisfies $K \geqslant 0$, then we have the inequality $k(s) \leqslant 2|H(\alpha(s))|$, where $H$ denotes the mean curvature of $S$ and $k$ the curvature of $\alpha$. Give an example where this inequality is sharp.

Paper 3, Section II, I

commentLet $S \subset \mathbb{R}^{N}$ be a manifold and let $\alpha:[a, b] \rightarrow S \subset \mathbb{R}^{N}$ be a smooth regular curve on $S$. Define the total length $L(\alpha)$ and the arc length parameter $s$. Show that $\alpha$ can be reparametrized by arc length.

Let $S \subset \mathbb{R}^{3}$ denote a regular surface, let $p, q \in S$ be distinct points and let $\alpha:[a, b] \rightarrow S$ be a smooth regular curve such that $\alpha(a)=p, \alpha(b)=q$. We say that $\alpha$ is length minimising if for all smooth regular curves $\tilde{\alpha}:[a, b] \rightarrow S$ with $\tilde{\alpha}(a)=p, \tilde{\alpha}(b)=q$, we have $L(\tilde{\alpha}) \geqslant L(\alpha)$. By deriving a formula for the derivative of the energy functional corresponding to a variation of $\alpha$, show that a length minimising curve is necessarily a geodesic. [You may use the following fact: given a smooth vector field $V(t)$ along $\alpha$ with $V(a)=V(b)=0$, there exists a variation $\alpha(s, t)$ of $\alpha$ such that $\left.\left.\partial_{s} \alpha(s, t)\right|_{s=0}=V(t) .\right]$

Let $\mathbb{S}^{2} \subset \mathbb{R}^{3}$ denote the unit sphere and let $S$ denote the surface $\mathbb{S}^{2} \backslash(0,0,1)$. For which pairs of points $p, q \in S$ does there exist a length minimising smooth regular curve $\alpha:[a, b] \rightarrow S$ with $\alpha(a)=p$ and $\alpha(b)=q$ ? Justify your answer.

Paper 4, Section II, I

commentLet $S \subset \mathbb{R}^{3}$ be a surface and $p \in S$. Define the exponential map exp $p$ and compute its differential $\left.d \exp _{p}\right|_{0}$. Deduce that $\exp _{p}$ is a local diffeomorphism.

Give an example of a surface $S$ and a point $p \in S$ for which the exponential map $\exp _{p}$ fails to be defined globally on $T_{p} S$. Can this failure be remedied by extending the surface? In other words, for any such $S$, is there always a surface $S \subset \widehat{S} \subset \mathbb{R}^{3}$ such that the exponential map $\widehat{\exp }_{p}$ defined with respect to $\widehat{S}$is globally defined on $T_{p} S=T_{p} \widehat{S}$?

State the version of the Gauss-Bonnet theorem with boundary term for a surface $S \subset \mathbb{R}^{3}$ and a closed disc $D \subset S$ whose boundary $\partial D$ can be parametrized as a smooth closed curve in $S$.

Let $S \subset \mathbb{R}^{3}$ be a flat surface, i.e. $K=0$. Can there exist a closed disc $D \subset S$, whose boundary $\partial D$ can be parametrized as a smooth closed curve, and a surface $\tilde{S} \subset \mathbb{R}^{3}$ such that all of the following hold:

(i) $(S \backslash D) \cup \partial D \subset \tilde{S}$;

(ii) letting $\tilde{D}$ be $(\tilde{S} \backslash(S \backslash D)) \cup \partial D$, we have that $\tilde{D}$ is a closed disc in $\tilde{S}$ with boundary $\partial \tilde{D}=\partial D$

(iii) the Gaussian curvature $\tilde{K}$ of $\tilde{S}$ satisfies $\tilde{K} \geqslant 0$, and there exists a $p \in \tilde{S}$ such that $\tilde{K}(p)>0$ ?

Justify your answer.

Paper 1, Section II, G

commentDefine what is meant by the regular values and critical values of a smooth map $f: X \rightarrow Y$ of manifolds. State the Preimage Theorem and Sard's Theorem.

Suppose now that $\operatorname{dim} X=\operatorname{dim} Y$. If $X$ is compact, prove that the set of regular values is open in $Y$, but show that this may not be the case if $X$ is non-compact.

Construct an example with $\operatorname{dim} X=\operatorname{dim} Y$ and $X$ compact for which the set of critical values is not a submanifold of $Y$.

[Hint: You may find it helpful to consider the case $X=S^{1}$ and $Y=\mathbf{R}$. Properties of bump functions and the function $e^{-1 / x^{2}}$ may be assumed in this question.]

Paper 2, Section II, G

commentIf an embedded surface $S \subset \mathbf{R}^{3}$ contains a line $L$, show that the Gaussian curvature is non-positive at each point of $L$. Give an example where the Gaussian curvature is zero at each point of $L$.

Consider the helicoid $S$ given as the image of $\mathbf{R}^{2}$ in $\mathbf{R}^{3}$ under the map

$\phi(u, v)=(\sinh v \cos u, \sinh v \sin u, u) .$

What is the image of the corresponding Gauss map? Show that the Gaussian curvature at a point $\phi(u, v) \in S$ is given by $-1 / \cosh ^{4} v$, and hence is strictly negative everywhere. Show moreover that there is a line in $S$ passing through any point of $S$.

[General results concerning the first and second fundamental forms on an oriented embedded surface $S \subset \mathbf{R}^{3}$ and the Gauss map may be used without proof in this question.]

Paper 3, Section II, G

commentExplain what it means for an embedded surface $S$ in $\mathbf{R}^{3}$ to be minimal. What is meant by an isothermal parametrization $\phi: U \rightarrow V \subset \mathbf{R}^{3}$ of an embedded surface $V \subset \mathbf{R}^{3}$ ? Prove that if $\phi$ is isothermal then $\phi(U)$ is minimal if and only if the components of $\phi$ are harmonic functions on $U$. [You may assume the formula for the mean curvature of a parametrized embedded surface,

$H=\frac{e G-2 f F+g E}{2\left(E G-F^{2}\right)}$

where $E, F, G$ (respectively $e, f, g$ ) are the coefficients of the first (respectively second) fundamental forms.]

Let $S$ be an embedded connected minimal surface in $\mathbf{R}^{3}$ which is closed as a subset of $\mathbf{R}^{3}$, and let $\Pi \subset \mathbf{R}^{3}$ be a plane which is disjoint from $S$. Assuming that local isothermal parametrizations always exist, show that if the Euclidean distance between $S$ and $\Pi$ is attained at some point $P \in S$, i.e. $d(P, \Pi)=\inf _{Q \in S} d(Q, \Pi)$, then $S$ is a plane parallel to $\Pi$.

Paper 4, Section II, G

commentFor $S \subset \mathbf{R}^{3}$ a smooth embedded surface, define what is meant by a geodesic curve on $S$. Show that any geodesic curve $\gamma(t)$ has constant speed $|\dot{\gamma}(t)|$.

For any point $P \in S$, show that there is a parametrization $\phi: U \rightarrow V$ of some open neighbourhood $V$ of $P$ in $S$, with $U \subset \mathbf{R}^{2}$ having coordinates $(u, v)$, for which the first fundamental form is

$d u^{2}+G(u, v) d v^{2}$

for some strictly positive smooth function $G$ on $U$. State a formula for the Gaussian curvature $K$ of $S$ in $V$ in terms of $G$. If $K \equiv 0$ on $V$, show that $G$ is a function of $v$ only, and that we may reparametrize so that the metric is locally of the form $d u^{2}+d w^{2}$, for appropriate local coordinates $(u, w)$.

[You may assume that for any $P \in S$ and nonzero $\xi \in T_{P} S$, there exists (for some $\epsilon>0)$ a unique geodesic $\gamma:(-\epsilon, \epsilon) \rightarrow S$ with $\gamma(0)=P$ and $\dot{\gamma}(0)=\xi$, and that such geodesics depend smoothly on the initial conditions $P$ and $\xi .]$

Paper 1, Section II, $22 G$

commentLet $\Omega \subset \mathbb{R}^{2}$ be a domain (connected open subset) with boundary $\partial \Omega$ a continuously differentiable simple closed curve. Denoting by $A(\Omega)$ the area of $\Omega$ and $l(\partial \Omega)$ the length of the curve $\partial \Omega$, state and prove the isoperimetric inequality relating $A(\Omega)$ and $l(\partial \Omega)$ with optimal constant, including the characterization for equality. [You may appeal to Wirtinger's inequality as long as you state it precisely.]

Does the result continue to hold if the boundary $\partial \Omega$ is allowed finitely many points at which it is not differentiable? Briefly justify your answer by giving either a counterexample or an indication of a proof.

Paper 2, Section II, G

commentIf $U$ denotes a domain in $\mathbb{R}^{2}$, what is meant by saying that a smooth map $\phi: U \rightarrow \mathbb{R}^{3}$ is an immersion? Define what it means for such an immersion to be isothermal. Explain what it means to say that an immersed surface is minimal.

Let $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ be an isothermal immersion. Show that it is minimal if and only if $x, y, z$ are harmonic functions of $u, v$. [You may use the formula for the mean curvature given in terms of the first and second fundamental forms, namely $\left.H=(e G-2 f F+g E) /\left(2\left\{E G-F^{2}\right\}\right) .\right]$

Produce an example of an immersed minimal surface which is not an open subset of a catenoid, helicoid, or a plane. Prove that your example does give an immersed minimal surface in $\mathbb{R}^{3}$.

Paper 3 , Section II, G

commentShow that the surface $S$ of revolution $x^{2}+y^{2}=\cosh ^{2} z$ in $\mathbb{R}^{3}$ is homeomorphic to a cylinder and has everywhere negative Gaussian curvature. Show moreover the existence of a closed geodesic on $S$.

Let $S \subset \mathbb{R}^{3}$ be an arbitrary embedded surface which is homeomorphic to a cylinder and has everywhere negative Gaussian curvature. By using a suitable version of the Gauss-Bonnet theorem, show that $S$ contains at most one closed geodesic. [If required, appropriate forms of the Jordan curve theorem in the plane may also be used without proof.

Paper 4, Section II, G

commentLet $\mathrm{U}(n)$ denote the set of $n \times n$ unitary complex matrices. Show that $\mathrm{U}(n)$ is a smooth (real) manifold, and find its dimension. [You may use any general results from the course provided they are stated correctly.] For $A$ any matrix in $\mathrm{U}(n)$ and $H$ an $n \times n$ complex matrix, determine when $H$ represents a tangent vector to $\mathrm{U}(n)$ at $A$.

Consider the tangent spaces to $\mathrm{U}(n)$ equipped with the metric induced from the standard (Euclidean) inner product $\langle\cdot, \cdot\rangle$ on the real vector space of $n \times n$ complex matrices, given by $\langle L, K\rangle=\operatorname{Re} \operatorname{trace}\left(L K^{*}\right)$, where $\operatorname{Re}$ denotes the real part and $K^{*}$ denotes the conjugate transpose of $K$. Suppose that $H$ represents a tangent vector to $\mathrm{U}(n)$ at the identity matrix $I$. Sketch an explicit construction of a geodesic curve on $\mathrm{U}(n)$ passing through $I$ and with tangent direction $H$, giving a brief proof that the acceleration of the curve is always orthogonal to the tangent space to $\mathrm{U}(n)$.

[Hint: You will find it easier to work directly with $n \times n$ complex matrices, rather than the corresponding $2 n \times 2 n$ real matrices.]

Paper 1, Section II, G

commentDefine the concepts of (smooth) manifold and manifold with boundary for subsets of $\mathbf{R}^{N}$.

Let $X \subset \mathbf{R}^{6}$ be the subset defined by the equations

$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=1, \quad x_{4}^{2}-x_{5}^{2}-x_{6}^{2}=-1 .$

Prove that $X$ is a manifold of dimension four.

For $a>0$, let $B(a) \subset \mathbf{R}^{6}$ denote the spherical ball $x_{1}^{2}+\ldots+x_{6}^{2} \leqslant a$. Prove that $X \cap B(a)$ is empty if $a<2$, is a manifold diffeomorphic to $S^{2} \times S^{1}$ if $a=2$, and is a manifold with boundary if $a>2$, with each component of the boundary diffeomorphic to $S^{2} \times S^{1}$.

[You may quote without proof any general results from lectures that you may need.]

Paper 2, Section II, G

commentDefine the terms Gaussian curvature $K$ and mean curvature $H$ for a smooth embedded oriented surface $S \subset \mathbf{R}^{3}$. [You may assume the fact that the derivative of the Gauss map is self-adjoint.] If $K=H^{2}$ at all points of $S$, show that both $H$ and $K$ are locally constant. [Hint: Use the symmetry of second partial derivatives of the field of unit normal vectors.]

If $K=H^{2}=0$ at all points of $S$, show that the unit normal vector $\mathbf{N}$ to $S$ is locally constant and that $S$ is locally contained in a plane. If $K=H^{2}$ is a strictly positive constant on $S$ and $\phi: U \rightarrow S$ is a local parametrization (where $U$ is connected) on $S$ with unit normal vector $\mathbf{N}(u, v)$ for $(u, v) \in U$, show that $\phi(u, v)+\mathbf{N}(u, v) / H$ is constant on $U$. Deduce that $S$ is locally contained in a sphere of radius $1 /|H|$.

If $S$ is connected with $K=H^{2}$ at all points of $S$, deduce that $S$ is contained in either a plane or a sphere.

Paper 3, Section II, G

commentLet $\alpha: I \rightarrow S$ be a parametrized curve on a smooth embedded surface $S \subset \mathbf{R}^{3}$. Define what is meant by a vector field $V$ along $\alpha$ and the concept of such a vector field being parallel. If $V$ and $W$ are both parallel vector fields along $\alpha$, show that the inner product $\langle V(t), W(t)\rangle$ is constant.

Given a local parametrization $\phi: U \rightarrow S$, define the Christoffel symbols $\Gamma_{j k}^{i}$ on $U$. Given a vector $v_{0} \in T_{\alpha(0)} S$, prove that there exists a unique parallel vector field $V(t)$ along $\alpha$ with $V(0)=v_{0}$ (recall that $V(t)$ is called the parallel transport of $v_{0}$ along $\alpha$ ).

Suppose now that the image of $\alpha$ also lies on another smooth embedded surface $S^{\prime} \subset \mathbf{R}^{3}$ and that $T_{\alpha(t)} S=T_{\alpha(t)} S^{\prime}$ for all $t \in I$. Show that parallel transport of a vector $v_{0}$ is the same whether calculated on $S$ or $S^{\prime}$. Suppose $S$ is the unit sphere in $\mathbf{R}^{3}$ with centre at the origin and let $\alpha:[0,2 \pi] \rightarrow S$ be the curve on $S$ given by

$\alpha(t)=(\sin \phi \cos t, \sin \phi \sin t, \cos \phi)$

for some fixed angle $\phi$. Suppose $v_{0} \in T_{P} S$ is the unit tangent vector to $\alpha$ at $P=\alpha(0)=$ $\alpha(2 \pi)$ and let $v_{1}$ be its image in $T_{P} S$ under parallel transport along $\alpha$. Show that the angle between $v_{0}$ and $v_{1}$ is $2 \pi \cos \phi$.

[Hint: You may find it useful to consider the circular cone $S^{\prime}$ which touches the sphere $S$ along the curve $\alpha$.]

Paper 4, Section II, G

commentLet $I=[0, l]$ be a closed interval, $k(s), \tau(s)$ smooth real valued functions on $I$ with $k$ strictly positive at all points, and $\mathbf{t}_{0}, \mathbf{n}_{0}, \mathbf{b}_{0}$ a positively oriented orthonormal triad of vectors in $\mathbf{R}^{3}$. An application of the fundamental theorem on the existence of solutions to ODEs implies that there exists a unique smooth family of triples of vectors $\mathbf{t}(s), \mathbf{n}(s), \mathbf{b}(s)$ for $s \in I$ satisfying the differential equations

$\mathbf{t}^{\prime}=k \mathbf{n}, \quad \mathbf{n}^{\prime}=-k \mathbf{t}-\tau \mathbf{b}, \quad \mathbf{b}^{\prime}=\tau \mathbf{n}$

with initial conditions $\mathbf{t}(0)=\mathbf{t}_{0}, \mathbf{n}(0)=\mathbf{n}_{0}$ and $\mathbf{b}(0)=\mathbf{b}_{0}$, and that $\{\mathbf{t}(s), \mathbf{n}(s), \mathbf{b}(s)\}$ forms a positively oriented orthonormal triad for all $s \in I$. Assuming this fact, consider $\alpha: I \rightarrow \mathbf{R}^{3}$ defined by $\alpha(s)=\int_{0}^{s} \mathbf{t}(t) d t$; show that $\alpha$ defines a smooth immersed curve parametrized by arc-length, which has curvature and torsion given by $k(s)$ and $\tau(s)$, and that $\alpha$ is uniquely determined by this property up to rigid motions of $\mathbf{R}^{3}$. Prove that $\alpha$ is a plane curve if and only if $\tau$ is identically zero.

If $a>0$, calculate the curvature and torsion of the smooth curve given by

$\alpha(s)=(a \cos (s / c), a \sin (s / c), b s / c), \quad \text { where } c=\sqrt{a^{2}+b^{2}}$

Suppose now that $\alpha:[0,2 \pi] \rightarrow \mathbf{R}^{3}$ is a smooth simple closed curve parametrized by arc-length with curvature everywhere positive. If both $k$ and $\tau$ are constant, show that $k=1$ and $\tau=0$. If $k$ is constant and $\tau$ is not identically zero, show that $k>1$. Explain what it means for $\alpha$ to be knotted; if $\alpha$ is knotted and $\tau$ is constant, show that $k(s)>2$ for some $s \in[0,2 \pi]$. [You may use standard results from the course if you state them precisely.]

Paper 1, Section II, H

commentFor $f: X \rightarrow Y$ a smooth map of manifolds, define the concepts of critical point, critical value and regular value.

With the obvious identification of $\mathbf{C}$ with $\mathbf{R}^{2}$, and hence also of $\mathbf{C}^{3}$ with $\mathbf{R}^{6}$, show that the complex-valued polynomial $z_{1}^{3}+z_{2}^{2}+z_{3}^{2}$ determines a smooth map $f: \mathbf{R}^{6} \rightarrow \mathbf{R}^{2}$ whose only critical point is at the origin. Hence deduce that $V:=f^{-1}((0,0)) \backslash\{\mathbf{0}\} \subset \mathbf{R}^{6}$ is a 4-dimensional manifold, and find the equations of its tangent space at any given point $\left(z_{1}, z_{2}, z_{3}\right) \in V$.

Now let $S^{5} \subset \mathbf{C}^{3}=\mathbf{R}^{6}$ be the unit 5 -sphere, defined by $\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}=1$. Given a point $P=\left(z_{1}, z_{2}, z_{3}\right) \in S^{5} \cap V$, by considering the vector $\left(2 z_{1}, 3 z_{2}, 3 z_{3}\right) \in \mathbf{C}^{3}=\mathbf{R}^{6}$ or otherwise, show that not all tangent vectors to $V$ at $P$ are tangent to $S^{5}$. Deduce that $S^{5} \cap V \subset \mathbf{R}^{6}$ is a compact three-dimensional manifold.

[Standard results may be quoted without proof if stated carefully.]

Paper 2, Section II, H

commentLet $\alpha:[0, L] \rightarrow \mathbf{R}^{3}$ be a regular curve parametrized by arc length having nowherevanishing curvature. State the Frenet relations between the tangent, normal and binormal vectors at a point, and their derivatives.

Let $S \subset \mathbf{R}^{3}$ be a smooth oriented surface. Define the Gauss map $N: S \rightarrow S^{2}$, and show that its derivative at $P \in S, d N_{P}: T_{P} S \rightarrow T_{P} S$, is self-adjoint. Define the Gaussian curvature of $S$ at $P$.

Now suppose that $\alpha:[0, L] \rightarrow \mathbf{R}^{3}$ has image in $S$ and that its normal curvature is zero for all $s \in[0, L]$. Show that the Gaussian curvature of $S$ at a point $P=\alpha(s)$ of the curve is $K(P)=-\tau(s)^{2}$, where $\tau(s)$ denotes the torsion of the curve.

If $S \subset \mathbf{R}^{3}$ is a standard embedded torus, show that there is a curve on $S$ for which the normal curvature vanishes and the Gaussian curvature of $S$ is zero at all points of the curve.

Paper 3, Section II, H

commentWe say that a parametrization $\phi: U \rightarrow S \subset \mathbf{R}^{3}$ of a smooth surface $S$ is isothermal if the coefficients of the first fundamental form satisfy $F=0$ and $E=G=\lambda(u, v)^{2}$, for some smooth non-vanishing function $\lambda$ on $U$. For an isothermal parametrization, prove that

$\phi_{u u}+\phi_{v v}=2 \lambda^{2} H \mathbf{N}$

where $\mathbf{N}$ denotes the unit normal vector and $H$ the mean curvature, which you may assume is given by the formula

$H=\frac{g+e}{2 \lambda^{2}}$

where $g=-\left\langle\mathbf{N}_{u}, \phi_{u}\right\rangle$ and $e=-\left\langle\mathbf{N}_{v}, \phi_{v}\right\rangle$ are coefficients in the second fundamental form.

Given a parametrization $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ of a surface $S \subset \mathbf{R}^{3}$, we consider the complex valued functions on $U$ :

$\theta_{1}=x_{u}-i x_{v}, \quad \theta_{2}=y_{u}-i y_{v}, \quad \theta_{3}=z_{u}-i z_{v}$

Show that $\phi$ is isothermal if and only if $\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}=0$. If $\phi$ is isothermal, show that $S$ is a minimal surface if and only if $\theta_{1}, \theta_{2}, \theta_{3}$ are holomorphic functions of the complex variable $\zeta=u+i v$

Consider the holomorphic functions on $D:=\mathbf{C} \backslash \mathbf{R}_{\geqslant 0}$ (with complex coordinate $\zeta=u+i v$ on $\mathbf{C})$ given by

$\theta_{1}:=\frac{1}{2}\left(1-\zeta^{-2}\right), \quad \theta_{2}:=-\frac{i}{2}\left(1+\zeta^{-2}\right), \quad \theta_{3}:=-\zeta^{-1}$

Find a smooth map $\phi(u, v)=(x(u, v), y(u, v), z(u, v)): D \rightarrow \mathbf{R}^{3}$ for which $\phi(-1,0)=\mathbf{0}$ and the $\theta_{i}$ defined by (2) satisfy the equations (1). Show furthermore that $\phi$ extends to a smooth map $\tilde{\phi}: \mathbf{C}^{*} \rightarrow \mathbf{R}^{3}$. If $w=x+i y$ is the complex coordinate on $\mathbf{C}$, show that

$\widetilde{\phi}(\exp (i w))=(\cosh y \cos x+1, \cosh y \sin x, y)$

Paper 4, Section II, H

Define what is meant by the geodesic curvature $k_{g}$ of a regular curve $\alpha: I \rightarrow S$ parametrized by arc length on a smooth oriented surface $S \subset \mathbf{R}^{3}$. If $S$ is the unit sphere in $\mathbf{R}^{3}$ and $\alpha: I \rightarrow S$ is a parametrized geodesic circle of radius $\phi$, with $0<\phi<\pi / 2$, justify the fact that $\left|k_{g}\right|=\cot \phi$.

State the general form of the Gauss-Bonnet theorem with boundary on an oriented surface $S$, explaining briefly the terms which occur.

Let $S \subset \mathbf{R}^{3}$ now denote the circular cone given by $z>0$ and $x^{2}+y^{2}=z^{2} \tan ^{2} \phi$, for a fixed choice of $\phi$ with $0<\phi<\pi / 2$, and with a fixed choice of orientation. Let $\alpha: I \rightarrow S$ be a simple closed piecewise regular curve on $S$, with (signed) exterior angles $\theta_{1}, \ldots, \theta_{N}$ at the vertices (that is, $\theta_{i}$