Geometry Of Surfaces

# Geometry Of Surfaces

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A2.7

comment(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.

A3.7

comment(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 $\mathbb{R}^{3}$.

A4.7

commentWrite an essay on the Gauss-Bonnet theorem and its proof.

A2.7

comment(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 $\lim _{R \rightarrow 0} a(R) / R^{4}=0$.

A3.7

comment(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 $\gamma$.]

A4.7

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

A $4 . 7 \quad$

commentWrite 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.

A2.7

comment(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".

A3.7

comment(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.

A $4 . 7 \quad$

commentWrite 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.

A2.7

comment(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.

A3.7

comment(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.