A2.7

Part II, 2001

(i) Give the definition of the curvature $\kappa(t)$ of a plane curve $\gamma:[a, b] \longrightarrow \mathbf{R}^{2}$ . Show that, if $\gamma:[a, b] \longrightarrow \mathbf{R}^{2}$ is a simple closed curve, then

\int_{a}^{b} \kappa(t)\|\dot{\gamma}(t)\| d t=2 \pi

(ii) Give the definition of a geodesic on a parametrized surface in $\mathbf{R}^{3}$ . Derive the differential equations characterizing geodesics. Show that a great circle on the unit sphere is a geodesic.

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A3.7

Geometry of Surfaces

Part II, 2001

(i) Give the definition of the surface area of a parametrized surface in $\mathbf{R}^{3}$ and show that it does not depend on the parametrization.

(ii) Let $\varphi(u)>0$ be a differentiable function of $u$ . Consider the surface of revolution:

\left(\begin{array}{l} u \\ v \end{array}\right) \mapsto f(u, v)=\left(\begin{array}{c} \varphi(u) \cos (v) \\ \varphi(u) \sin (v) \\ u \end{array}\right)

Find a formula for each of the following: (a) The first fundamental form. (b) The unit normal. (c) The second fundamental form. (d) The Gaussian curvature.

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A $4 . 7 \quad$

Geometry of Surfaces

Part II, 2001

Write an essay on the Gauss-Bonnet theorem. Make sure that your essay contains a precise statement of the theorem, in its local form, and a discussion of some of its applications, including the global Gauss-Bonnet theorem.

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A2.7

Geometry of Surfaces

Part II, 2002

(i)

Consider the surface

z=\frac{1}{2}\left(\lambda x^{2}+\mu y^{2}\right)+h(x, y)

where $h(x, y)$ is a term of order at least 3 in $x, y$ . Calculate the first fundamental form at $x=y=0$ .

(ii) Calculate the second fundamental form, at $x=y=0$ , of the surface given in Part (i). Calculate the Gaussian curvature. Explain why your answer is consistent with Gauss' "Theorema Egregium".

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A3.7

Geometry of Surfaces

Part II, 2002

(i) State what it means for surfaces $f: U \rightarrow \mathbb{R}^{3}$ and $g: V \rightarrow \mathbb{R}^{3}$ to be isometric.

Let $f: U \rightarrow \mathbb{R}^{3}$ be a surface, $\phi: V \rightarrow U$ a diffeomorphism, and let $g=f \circ \phi: V \rightarrow$ $\mathbb{R}^{3} .$

State a formula comparing the first fundamental forms of $f$ and $g$ .

(ii) Give a proof of the formula referred to at the end of part (i). Deduce that "isometry" is an equivalence relation.

The catenoid and the helicoid are the surfaces defined by

(u, v) \rightarrow(u \cos v, u \sin v, v)

and

(\vartheta, z) \rightarrow(\cosh z \cos \vartheta, \cosh z \sin \vartheta, z)

Show that the catenoid and the helicoid are isometric.

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A $4 . 7 \quad$

Geometry of Surfaces

Part II, 2002

Write an essay on the Euler number of topological surfaces. Your essay should include a definition of subdivision, some examples of surfaces and their Euler numbers, and a discussion of the statement and significance of the Gauss-Bonnet theorem.

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A2.7

Geometry of Surfaces

Part II, 2003

(i) What are geodesic polar coordinates at a point $P$ on a surface $M$ with a Riemannian metric $d s^{2}$ ?

Assume that

d s^{2}=d r^{2}+H(r, \theta)^{2} d \theta^{2}

for geodesic polar coordinates $r, \theta$ and some function $H$ . What can you say about $H$ and $d H / d r$ at $r=0$ ?

(ii) Given that the Gaussian curvature $K$ may be computed by the formula $K=-H^{-1} \partial^{2} H / \partial r^{2}$ , show that for small $R$ the area of the geodesic disc of radius $R$ centred at $P$ is

\pi R^{2}-(\pi / 12) K R^{4}+a(R),

where $a(R)$ is a function satisfying $\lim _{R \rightarrow 0} a(R) / R^{4}=0$ .

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A3.7

Geometry of Surfaces

Part II, 2003

(i) Suppose that $C$ is a curve in the Euclidean $(\xi, \eta)$ -plane and that $C$ is parameterized by its arc length $\sigma$ . Suppose that $S$ in Euclidean $\mathbb{R}^{3}$ is the surface of revolution obtained by rotating $C$ about the $\xi$ -axis. Take $\sigma, \theta$ as coordinates on $S$ , where $\theta$ is the angle of rotation.

Show that the Riemannian metric on $S$ induced from the Euclidean metric on $\mathbb{R}^{3}$ is

d s^{2}=d \sigma^{2}+\eta(\sigma)^{2} d \theta^{2}

(ii) For the surface $S$ described in Part (i), let $e_{\sigma}=\partial / \partial \sigma$ and $e_{\theta}=\partial / \partial \theta$ . Show that, along any geodesic $\gamma$ on $S$ , the quantity $g\left(\dot{\gamma}, e_{\theta}\right)$ is constant. Here $g$ is the metric tensor on $S$ .

[You may wish to compute $\left[X, e_{\theta}\right]=X e_{\theta}-e_{\theta} X$ for any vector field $X=A e_{\sigma}+B e_{\theta}$ , where $A, B$ are functions of $\sigma, \theta$ . Then use symmetry to compute $D_{\dot{\gamma}}\left(g\left(\dot{\gamma}, e_{\theta}\right)\right)$ , which is the rate of change of $g\left(\dot{\gamma}, e_{\theta}\right)$ along $\gamma$ .]

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A4.7

Geometry of Surfaces

Part II, 2003

Write an essay on the Theorema Egregium for surfaces in $\mathbb{R}^{3}$ .

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A2.7

Geometry of Surfaces

Part II, 2004

(i) What is a geodesic on a surface $M$ with Riemannian metric, and what are geodesic polar co-ordinates centred at a point $P$ on $M$ ? State, without proof, formulae for the Riemannian metric and the Gaussian curvature in terms of geodesic polar co-ordinates.

(ii) Show that a surface with constant Gaussian curvature 0 is locally isometric to the Euclidean plane.

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A3.7

Geometry of Surfaces

Part II, 2004

(i) The catenoid is the surface $C$ in Euclidean $\mathbb{R}^{3}$ , with co-ordinates $x, y, z$ and Riemannian metric $d s^{2}=d x^{2}+d y^{2}+d z^{2}$ obtained by rotating the curve $y=\cosh x$ about the $x$ -axis, while the helicoid is the surface $H$ swept out by a line which lies along the $x$ -axis at time $t=0$ , and at time $t=t_{0}$ is perpendicular to the $z$ -axis, passes through the point $\left(0,0, t_{0}\right)$ and makes an angle $t_{0}$ with the $x$ -axis.

Find co-ordinates on each of $C$ and $H$ and write $x, y, z$ in terms of these co-ordinates.

(ii) Compute the induced Riemannian metrics on $C$ and $H$ in terms of suitable coordinates. Show that $H$ and $C$ are locally isometric. By considering the $x$ -axis in $H$ , show that this local isometry cannot be extended to a rigid motion of any open subset of Euclidean $\mathbb{R}^{3}$ .