Math 120A: Differential Geometry

Part I: Curves in space

Part II: Global theory of plane curves

Part III: Surfaces and the first fundamental form

Part IV: Curves on surfaces, geodesics, parallel transport

Part V: Curvature of surfaces

Homework 1

Covers Lectures 1 to 2. Quiz 1 is based on these problems.

1. Let $\alpha(\theta) = (e^\theta\cos\theta, e^\theta\sin\theta, 0)$. Prove that the angle between $\alpha$ and $T$ is constant.

2. Let $\alpha(t)$ be a regular curve. Suppose there is a point $a \in \mathbb{R}^3$ such that $\alpha(t) - a$ is orthogonal to $T(t)$ for all $t$. Prove that $\alpha(t)$ lies on a sphere. (Hint: what is the center of the sphere?)

3. Consider the function $\alpha \colon \mathbb{R} \to \mathbb{R}^3$ given by $\alpha(t) = (t^2, t^3, 0)$. Prove that $\alpha$ is $C^1$ but not regular. Show that the image of $\alpha$ has a sharp corner by graphing $\alpha$.

4. Let $g \colon (0,\infty) \to (0,1)$ be given by $g(r) = r^2/(r^2+1)$. Is this a reparametrization?

5. Find the arc length of the circular helix $\alpha(t) = (r\cos t, r\sin t, ht)$, $0 \le t \le 10$.

6. Reparametrize the circular helix $\alpha(t) = (r\cos t, r\sin t, ht)$ by arc length.

7. Find the arc length of $\alpha(t) = (2\cosh 3t, -2\sinh 3t, 6t)$, $0 \le t \le 5$.

8. Reparametrize the curve $\alpha(t) = (e^t\cos t, e^t\sin t, e^t)$ by arc length.

9. Prove that the following is a unit speed curve. $$\alpha(s) = \tfrac{1}{2}\Big(s + \sqrt{s^2+1},\ \big(s + \sqrt{s^2+1}\big)^{-1},\ \sqrt{2}\,\ln\big(s + \sqrt{s^2+1}\big)\Big).$$

10. Let $\alpha(t)$ be a regular curve with $|d\alpha/dt| = a$, where $a$ is a fixed positive constant. If $s$ is the arc length measured from some point, prove that $t = c + s/a$ for some constant $c$.

11. Let $\alpha(t) = (\sin 3t\cos t,\ \sin 3t\sin t,\ 0)$. Show that $\alpha$ is a regular curve, and find the equation of its tangent line at $t = \pi/3$.

12. Which of the following are regular curves? (a) $\alpha(\theta) = (\cos\theta,\ 1 - \cos\theta - \sin\theta,\ -\sin\theta)$; (b) $\beta(\theta) = (2\sin^2\theta,\ 2\sin^2\theta\tan\theta,\ 0)$, $-\pi/2 < \theta < \pi/2$; (c) $\gamma(\theta) = (\cos\theta,\ \cos^2\theta,\ \sin\theta)$. For each regular one, find the tangent line at $\theta = \pi/4$.

13. Show that $f \colon (-1, 1) \to \mathbb{R}$, $f(t) = \tan(\pi t/2)$, is a reparametrization.

14. Reparametrize the curve $\alpha(t) = (\cosh t,\ \sinh t,\ t)$ by arc length.

15. Find the length of one arch of the cycloid $\alpha(t) = (t - \sin t,\ 1 - \cos t,\ 0)$, $0 \le t \le 2\pi$.

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Homework 2

Covers Lectures 3 to 5. Quiz 2 is based on these problems.

1. Show that $\alpha(s) = \tfrac{1}{13}\big(5\cos s,\ 8 - 13\sin s,\ -12\cos s\big)$ is a unit speed curve and compute its Frenet-Serret apparatus.

2. Show that $\alpha(s) = \Big(\frac{(1+s)^{3/2}}{3},\ \frac{(1-s)^{3/2}}{3},\ \frac{s}{\sqrt{2}}\Big)$ is a unit speed curve and compute its Frenet-Serret apparatus.

3. Let $\alpha(s)$ be a curve in the $(x,y)$-plane. Prove that if $\kappa \ne 0$ then $\tau = 0$. (Hint: first prove $B = \pm(0,0,1)$.)

4. Let $\alpha(s) = (x(s), y(s), 0)$ be a unit speed curve. Prove that $\kappa = |x'y'' - x''y'|$.

5. Let $\alpha(s)$ be a unit speed curve with $\kappa \ne 0$. Prove that $$\tau = \frac{[\alpha', \alpha'', \alpha''']}{\langle \alpha'', \alpha'' \rangle},$$ where $[u,v,w] = \langle u \times v, w \rangle$ is the triple product.

6. Let $\alpha(s)$ be a unit speed plane curve ($\tau = 0$) with constant curvature $\kappa > 0$. Integrate the Frenet-Serret equations directly to show that $\alpha$ is a circle of radius $1/\kappa$.

7. Let $\alpha(s)$ be a unit speed curve with $\kappa \ne 0$. Find a vector $w(s)$ such that $T' = w \times T$, $N' = w \times N$, $B' = w \times B$.

8. Let $\alpha(s)$ be a unit speed curve with constant $\kappa > 0$ and constant $\tau$. Prove that $\alpha$ is a circular helix by showing that its projection onto a plane perpendicular to $\omega = \tau T + \kappa B$ is a circle.

9. Let $f(t) = e^{-1/t^2}$ if $t \ne 0$ and $f(0) = 0$. You may assume $f$ is $C^\infty$ with all derivatives $0$ at $0$. Let $\alpha(t) = (t, f(t), 0)$ if $t \le 0$ and $\alpha(t) = (t, 0, f(t))$ if $t > 0$. Prove that $\alpha$ is regular and $C^\infty$. Show that $\kappa = 0$ at $t = 0$. (Note: $t$ is not the arc length.)

10. Find a unit speed curve $\alpha(s)$ with $\kappa(s) = 1/(1+s^2)$ and $\tau = 0$.

11. Show that $\alpha(s) = \tfrac12\big(\cos^{-1}s - s\sqrt{1 - s^2},\ 1 - s^2,\ 0\big)$, $-1 < s < 1$, is a unit speed curve and compute its Frenet-Serret apparatus.

12. Let $\alpha(s)$ be a unit speed curve with $\kappa \ne 0$. Prove that $\kappa\tau = -\langle T', B' \rangle$.

13. Find the equations of the osculating planes of the curves in Problems 1 and 2 at $s = 0$.

14. Let $\alpha(s)$ be a unit speed curve with $\kappa \ne 0$ such that all of its normal lines $\{\alpha(s) + \lambda N(s) : \lambda \in \mathbb{R}\}$ pass through a fixed point $x_0$. Prove that $\alpha$ is part of a circle.

15. Find the unit speed plane curve $\alpha(s)$, $s > 0$, with $\kappa(s) = 1/s$ and $\tau = 0$, such that $\alpha(1) = (0, 0, 0)$ and $T(1) = (1, 0, 0)$.

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Homework 3

Covers Lectures 6 to 9. Quiz 3 is based on these problems.

1. Compute the Frenet-Serret apparatus for the non-unit speed curve $\beta(t) = (t, t^2, t^3)$.

2. Find the curvature and torsion of the non-unit speed curve $(e^t\cos t, e^t\sin t, e^t)$.

3. Compute $\int_C (x\,dy - y\,dx)/(x^2+y^2)$, where $C$ is the straight line segment (a) from $(0,1)$ to $(1,0)$; (b) from $(1,0)$ to $(1,1)$; (c) from $(1,1)$ to $(0,1)$.

4. Verify Green's theorem for $\int_C (x\,dy - y\,dx)/(x^2+y^2)$, where $C$ is the triangle with vertices $(0,1)$, $(1,0)$, $(1,1)$.

5. For a unit speed plane curve $\alpha$, prove that $\kappa(s) = |k(s)|$, that if $\kappa(s) \ne 0$ then $n(s) = \pm N(s)$, and that $n'(s) = -k(s)T(s)$, where $k$ is the plane curvature.

6. If $\alpha$ is a unit speed closed plane curve with period $L$, prove that there is a well-defined function $\theta \colon [0,L] \to \mathbb{R}$ such that $T(s) = (\cos\theta(s), \sin\theta(s))$.

7. If $\alpha$ is a simple closed unit speed plane curve, prove that the tangent direction $T \colon [0,L] \to S^1$ is onto.

8. Let $C$ be the closed curve made up of the straight line segment from $(-1,0)$ to $(1,0)$ and two circular arcs of radius $2$, one centered at $(-1,0)$ from $(1,0)$ to $(0,\sqrt{3})$ and one centered at $(1,0)$ from $(0,\sqrt{3})$ to $(-1,0)$. Compute the integrals of $k$ along the smooth pieces and the exterior angles at the corners, and verify that the rotation index is $1$.

9. Let $\alpha$ be a simple closed plane curve, traversed counterclockwise, bounding a region of area $A$. Use Green's theorem to prove that $A = \int_\alpha x\,dy = -\int_\alpha y\,dx$.

10. Let $a$ and $b$ be positive numbers. Prove that $\sqrt{ab} \le \tfrac12(a+b)$, with equality if and only if $a = b$. (This is used in the proof of the isoperimetric inequality.)

11. Compute the Frenet-Serret apparatus of the non-unit speed curve $\beta(t) = (t - \cos t,\ \sin t,\ t)$.

12. Prove that a regular curve $\beta(t)$ is a straight line if and only if $\dot\beta$ and $\ddot\beta$ are linearly dependent for all $t$.

13. Let $\beta(t) = (x(t), y(t))$ be a regular plane curve, not necessarily unit speed. Prove that its signed curvature is $k = \dfrac{\dot x\ddot y - \ddot x\dot y}{(\dot x^2 + \dot y^2)^{3/2}}$. Compute $k$ for the ellipse $(a\cos t, b\sin t)$, $a, b > 0$, and find where it is largest.

14. (a) If $\alpha(s)$ is a closed unit speed plane curve and $\beta(s) = \alpha(-s)$, show that the rotation index of $\beta$ is the negative of that of $\alpha$. (b) Sketch the curve $\gamma(t) = (\sin t,\ \sin 2t)$, $0 \le t \le 2\pi$, and determine its rotation index, justifying your answer.

15. Let $P_m$ be a regular polygon with $m$ sides of length $1$. Compute $L^2/(4\pi A)$ for $P_m$, verify that it is greater than $1$, and show that it tends to $1$ as $m \to \infty$.

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Homework 4

Covers Lectures 10 to 11. Quiz 4 is based on these problems.

1. Define $$x(u^1,u^2) = \big((2+\cos u^1)\cos u^2,\ (2+\cos u^1)\sin u^2,\ \sin u^1\big)$$ with $-\pi < u^1, u^2 < \pi$. Prove that $x$ is a simple surface. Compute $x_1$, $x_2$, $n$ as functions of $u^1, u^2$.

2. Let $\alpha(t) = (r(t), z(t))$ be a regular one-one curve with $r(t) > 0$ for all $t$, and consider $$x(t,\theta) = (r(t)\cos\theta,\ r(t)\sin\theta,\ z(t)),\qquad -\pi < \theta < \pi.$$ Prove that this is a simple surface. Compute $x_1$, $x_2$, $n$.

3. The surface from Problem 2 is called a surface of revolution; the $t$-curves are called meridians, the $\theta$-curves circles of latitude, and the $z$-axis the axis of revolution. Prove that the surface from Problem 1 is a surface of revolution, and find its meridians and circles of latitude.

4. Let $U = \{(u^1,u^2) : 0 < u^1,\ 0 < u^2 < 2\pi\}$ and $x(u^1,u^2) = (u^1\cos u^2,\ u^1\sin u^2,\ u^1 + u^2)$. Prove that $x$ is a simple surface.

5. Let $\theta(u^1)$ be a $C^k$ function. Prove that $x(u^1,u^2) = (u^2\cos\theta(u^1),\ u^2\sin\theta(u^1),\ u^1)$ is a simple surface.

6. Let $x(u^1,u^2) = (u^1+u^2,\ u^1-u^2,\ u^1u^2)$. Show that $x$ is a simple surface. Find the normal and the equation of the tangent plane at $u^1 = 1$, $u^2 = 2$.

7. Let $$x(\theta,v) = (\cos\theta, \sin\theta, 0) + v\big(\sin\tfrac{\theta}{2}\cos\theta,\ \sin\tfrac{\theta}{2}\sin\theta,\ \cos\tfrac{\theta}{2}\big)$$ with $\theta \in (-\pi,\pi)$, $v \in (-\tfrac12, \tfrac12)$. Compute $n(\theta,0)$ and show that $$\lim_{\theta\to-\pi} n(\theta,0) = -\lim_{\theta\to\pi} n(\theta,0), \qquad\text{while}\qquad \lim_{\theta\to-\pi} x(\theta,0) = \lim_{\theta\to\pi} x(\theta,0).$$ Draw the surface.

8. Let $S^2 = \{(u,v,w) \in \mathbb{R}^3 : u^2+v^2+w^2 = 1\}$ and $\mathbb{R}^2 = \{(u,v,w) \in \mathbb{R}^3 : w = 0\}$. The line joining $(u,v,0)$ and $(0,0,1)$ intersects $S^2$ in a point other than $(0,0,1)$; call this point $x(u,v)$, and prove that $x \colon \mathbb{R}^2 \to \mathbb{R}^3$ is a simple surface. (The inverse map is called stereographic projection.)

9. Let $x(\theta,\phi) = (\cos\phi\cos\theta,\ 2\cos\phi\sin\theta,\ 3\sin\phi)$, $\theta \in (-\pi,\pi)$, $\phi \in (-\pi/2, \pi/2)$. Show that $x$ is a simple surface. What is it?

10. Let $x(u,v) = (\sqrt{1-u^2}\cos v,\ \sqrt{1-u^2}\sin v,\ u)$, $u \in (-1,1)$, $v \in (-\pi,\pi)$. Show that $x$ is a simple surface. What is it?

11. Let $x(r, s) = (r - s,\ r + s,\ 2(r^2 + s^2))$ for all $(r, s) \in \mathbb{R}^2$. Show that $x$ is a simple surface. Can you describe the surface geometrically?

12. Let $\alpha(s)$ be a unit speed curve with $\kappa \ne 0$, and let $x(s, t) = \alpha(s) + t\,\alpha'(s)$ for $t \ne 0$. Show that $x$ satisfies the regularity condition $x_1 \times x_2 \ne 0$. (Where $x$ is one-to-one, it is called the tangent developable surface of $\alpha$.)

13. Show that the helicoid $x(u, v) = (u\cos v,\ u\sin v,\ v)$, $(u, v) \in \mathbb{R}^2$, is a simple surface, and find its unit normal and the equation of its tangent plane at $(u, v) = (1, \pi/2)$.

14. Let $x(u^1, u^2)$ be a coordinate patch on an open set $U$, let $V = \{(v^1, v^2) : (v^2, v^1) \in U\}$, and let $f(v^1, v^2) = (v^2, v^1)$. Show that $f$ is a coordinate transformation, that $y = x \circ f$ is a coordinate patch, and that $y$ has the opposite unit normal to $x$.

15. At the point $P = (\tfrac12, \tfrac12, \tfrac{\sqrt2}{2})$ of the unit sphere, write the vectors $x_\theta$ and $x_\phi$ of spherical coordinates $x(\theta, \phi) = (\cos\theta\cos\phi, \sin\theta\cos\phi, \sin\phi)$ as linear combinations of the vectors $y_1$, $y_2$ of the Monge patch $y(u^1, u^2) = (u^1, u^2, \sqrt{1 - (u^1)^2 - (u^2)^2})$.

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Homework 5

Covers Lectures 12 to 13. Quiz 5 is based on these problems.

1. The coordinate patch from Homework 4, Problem 2 does not cover the entire surface of revolution: it omits the points that would correspond to $\theta = \pm\pi$. Define a second coordinate patch with $0 < \theta < 2\pi$, check the overlap condition, and thus show that the surface of revolution is a surface.

2. Homework 4, Problem 1 gives a coordinate patch for a torus (the surface of a doughnut). Find three more patches making the entire torus a surface.

3. Every point of $S^2$ is either in the image of the patch from Homework 4, Problem 8, or in the image of the patch $x(u^1,u^2) = \big(u^1, u^2, \sqrt{1-(u^1)^2-(u^2)^2}\big)$, $(u^1)^2 + (u^2)^2 < 1$. Show that these two patches make $S^2$ into a surface by computing the coordinate transformation on the overlap.

4. Describe some possible parametrizations of the ellipsoid $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$.

5. Give another coordinate patch for the surface from Homework 4, Problem 7, so that the points with $\theta = \pm\pi$ are included. Note that Homework 4, Problem 7 implies that it is impossible to choose a unit normal on this entire surface continuously.

6. Prove that the metric coefficients of the surface of revolution from Homework 4, Problem 2 are given by $$\begin{pmatrix} \dot r^2 + \dot z^2 & 0 \\ 0 & r^2 \end{pmatrix}.$$

7. Compute the metric coefficients of the parametrization of $S^2$ from Homework 4, Problem 8.

8. Compute the inverses of the matrices of metric coefficients from Problems 6 and 7.

9. A coordinate patch is called an orthogonal net if the $u^1$-curves meet the $u^2$-curves at right angles. Prove that the meridians and circles of latitude of a surface of revolution form an orthogonal net.

10. Let $x, y$ be the Cartesian coordinates and $r, \theta$ the polar coordinates on the plane $\mathbb{R}^2$. Prove that $x = r\cos\theta$, $y = r\sin\theta$ is a coordinate transformation. Show that the metric coefficients of the plane with respect to Cartesian coordinates form the identity matrix, and compute the metric coefficients of the plane in polar coordinates.

11. Show that the hyperboloid of one sheet $x^2 + y^2 - z^2 = 1$ is a surface by covering it with patches of the form $x(\theta, t) = (\cosh t\cos\theta,\ \cosh t\sin\theta,\ \sinh t)$ and checking the overlap condition.

12. Show that the half-cone $\{(x, y, z) : x^2 + y^2 = z^2,\ z > 0\}$ is a surface.

13. For a coordinate patch $x$, show that $u^1$ is arc length along the $u^1$-curves if and only if $g_{11} \equiv 1$.

14. Consider the patch $x(r, \theta) = (r\cos\theta,\ r\sin\theta,\ r)$, $r > 0$, and the curve $\gamma(t) = x(r(t), \theta(t))$ with $r(t) = e^{t\cot\beta/\sqrt2}$ and $\theta(t) = t/\sqrt2$, where $0 < \beta < \pi/2$ is a constant. Find the length of $\gamma$ for $0 \le t \le \pi$, and show that $\beta$ is the angle between $\gamma$ and the lines $\theta = $ constant.

15. Find the angle between the parameter curves of the saddle $x(u, v) = (u, v, uv)$ at the point $(u, v) = (1, 1)$.

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Homework 6

Covers Lectures 14 to 15. Quiz 6 is based on these problems.

1. Prove that $\kappa^2 = \kappa_n^2 + \kappa_g^2$.

2. Show that the matrix $(L_{ij})$ for the surface of revolution from Homework 4, Problem 2 is $$\frac{1}{\sqrt{\dot r^2 + \dot z^2}} \begin{pmatrix} \dot r\ddot z - \ddot r\dot z & 0 \\ 0 & r\dot z \end{pmatrix}.$$

3. Prove that for a surface of revolution, $\det(L_{ij}) \equiv 0$ if and only if each meridian is a straight line.

4. Let $\overline{L}_{\alpha\beta}$ be the corresponding expression for $L_{ij}$ in a coordinate system $V$, and let $f \colon U \to V$ be a coordinate transformation. Prove that $$L_{ij} = \pm\sum_{\alpha,\beta} \overline{L}_{\alpha\beta}\,\frac{\partial v^\alpha}{\partial u^i}\,\frac{\partial v^\beta}{\partial u^j}.$$

5. For the patch $x(u,v) = (u, v, u^2+v^2)$, find the normal curvature of the curve $\gamma(t) = x(t^2, t)$ at $t = 1$.

6. Consider the plane as the simple surface $x(r,s) = (r, s, 0)$. Prove that the geodesic curvature of a curve in the plane is its plane curvature.

7. The sphere is a surface of revolution. Find the geodesic curvature of a circle of latitude.

8. Let $\gamma$ be a curve on a sphere. Prove that $\kappa_n$ is constant.

9. Let $\gamma$ be a curve on a sphere with $\kappa_g$ constant. Prove that $\gamma$ is a circle.

10. Does the sign of the geodesic curvature have any meaning in a coordinate patch? In a surface?

11. Let $g = \det(g_{ij})$. Prove that $\dfrac12\dfrac{\partial(\ln g)}{\partial u^1} = \Gamma^1_{11} + \Gamma^2_{12}$ and $\dfrac12\dfrac{\partial(\ln g)}{\partial u^2} = \Gamma^1_{21} + \Gamma^2_{22}$.

12. Compute all the Christoffel symbols of the plane in polar coordinates $x(r, \theta) = (r\cos\theta, r\sin\theta, 0)$.

13. For the surface of revolution of Homework 4, Problem 2, suppose that the curve $(r(t), z(t))$ has unit speed. Compute all the Christoffel symbols.

14. On the saddle $x(u, v) = (u, v, uv)$, consider the curve $\gamma(t) = x(t, t)$. Find its normal curvature and geodesic curvature at $t = 0$.

15. Prove that a unit speed curve on a surface has $\kappa_n \equiv 0$ and $\kappa_g \equiv 0$ if and only if it is a straight line.

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Homework 7

Covers Lectures 16 to 18. Quiz 7 is based on these problems.

1. Prove that the meridians of a surface of revolution are geodesics. Also determine which circles of latitude are geodesics.

2. Let $M$ be a surface and $\Pi$ a plane that intersects $M$ in a curve $\gamma$. Show that $\gamma$ is a geodesic if $\Pi$ is a plane of symmetry of $M$, that is, if the two sides of $\Pi$ are mirror images.

3. Let $\gamma(t)$ be a regular curve on a surface $M$ whose velocity $d\gamma/dt$ is a parallel vector field along $\gamma$. Prove that there are constants $a \ne 0$ and $b$ such that $s = at + b$, where $s$ is arc length, and conclude that $\gamma$ is a geodesic up to reparametrization.

4. Suppose $x$ is a coordinate patch with $g_{11} \equiv 1$ and $g_{12} \equiv 0$. Prove that the $u^1$-curves are geodesics.

5. If $M$ is a surface of revolution and $\gamma$ is a geodesic, prove that $r\cos\beta(s)$ is constant, where $\beta(s)$ is the angle between $\gamma'(s)$ and the circle of latitude (of radius $r$) through $\gamma(s)$.

6. Let $\gamma(t)$ be a geodesic that is not parametrized by arc length. Prove that, for each $i = 1, 2$, $$\frac{d^2\gamma^i}{dt^2} + \sum_{j,k}\Gamma^i_{jk}\frac{d\gamma^j}{dt}\frac{d\gamma^k}{dt} = -\frac{d\gamma^i}{dt}\,\frac{d^2t}{ds^2}\Big(\frac{ds}{dt}\Big)^2.$$

7. Consider the patch $x(u,v) = (u, v, uv)$. Show that the non-unit speed curve $\gamma(t) = (t, -t, -t^2)$ is a geodesic when parametrized by arc length.

8. Let $\alpha(s) = (f(s), g(s))$ be a simple unit speed plane curve, and let $x(s,t) = (f(s), g(s), t)$. Let $\beta$ be a fixed constant, and let $\gamma(\theta) = (f(\theta), g(\theta), \theta\tan\beta)$. Prove that $\gamma$ is a geodesic. (Note: $\theta$ is not the arc length.)

9. Let $\gamma(s)$ be a unit speed curve on a surface $M$, and let $S = n \times T$. Prove that $S$ is parallel along $\gamma$ if and only if $\gamma$ is a geodesic.

10. Let $\gamma$ be a curve on a surface $M$ from $P$ to $Q$. For $X \in T_PM$, let $\gamma^\parallel(X) \in T_QM$ be the value at $Q$ of the parallel translate of $X$ along $\gamma$. Prove that $\gamma^\parallel \colon T_PM \to T_QM$ is a linear isomorphism and an isometry: $\langle \gamma^\parallel(X), \gamma^\parallel(Y) \rangle = \langle X, Y \rangle$.

11. Using Homework 6, Problem 12, show that a curve $(r(s), \theta(s))$ in the plane, in polar coordinates, satisfies the geodesic equations if and only if $r'' - r(\theta')^2 = 0$ and $\theta'' + \frac{2}{r}r'\theta' = 0$. Check that the line $r\cos\theta = 1$, suitably parametrized, is a solution.

12. Find as many geodesics as you can on the hyperboloid $x^2 + y^2 - z^2 = 1$.

13. Let $M$ be the unit sphere with the north pole $(0, 0, 1)$ removed, and let $0 < a < \pi/2$. Let $P = (\sin a, 0, \cos a)$ and $Q = (-\sin a, 0, \cos a)$. Show that there is a geodesic in $M$ from $P$ to $Q$, but no shortest curve in $M$ from $P$ to $Q$.

14. On the cone $x(r, \theta) = (r\cos\theta, r\sin\theta, r)$, with Christoffel symbols $\Gamma^1_{22} = -r/2$, $\Gamma^2_{12} = \Gamma^2_{21} = 1/r$ (the others zero), solve the parallel transport equations along the circle $r = r_0$, $\theta = t$. Show that after going once around ($t$ from $0$ to $2\pi$), a vector has turned by the angle $\sqrt2\,\pi$ relative to the frame $x_r, x_\theta/r_0$.

15. Let $\gamma$ be a unit speed curve on a surface, and let $X_N$ be the tangential component of its principal normal $N$. (a) Prove that $X_N = N - \langle N, n \rangle n$ and that $X_N$ is a vector field along $\gamma$. (b) Prove that the following are equivalent: (i) $X_N = 0$; (ii) $\gamma$ is a geodesic; (iii) $X_N$ is parallel along $\gamma$.

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Homework 8

Covers Lectures 19 to 21. Quiz 8 is based on these problems.

1. Show that for $M = S^2$ the Weingarten map is $L = \pm\mathrm{Id}$, by computing $L_{ik}$ in a coordinate patch and raising an index.

2. Show that for $M = S^1 \times (0,1)$, $L$ can be represented by the matrix $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$.

3. Find $L$ for the torus from Homework 4, Problem 1.

4. Prove that $\mathrm{II}(X,Y) = \langle L(X), Y \rangle = \langle X, L(Y) \rangle$ for all $X, Y \in T_pM$.

5. Find the matrix $(L^i_j)$ for a surface of revolution.

6. Let $f \colon V \to U$ be a change of coordinates. How are the $\overline{L}^\alpha_\beta$ related to the $L^i_j$?

7. Let $\gamma(s) = x(\gamma^1(s), \gamma^2(s))$ be a unit speed curve on a surface, so that $(T, S, n)$ is a positive orthonormal frame, with $n(s) = n(\gamma^1(s), \gamma^2(s))$ viewed as a function of $s$. Prove the following analogues of the Frenet-Serret equations: $$T' = \mathrm{II}(T,T)\,n + \kappa_g S,\qquad S' = -\kappa_g T + \mathrm{II}(T,S)\,n,\qquad n' = -\mathrm{II}(T,T)\,T - \mathrm{II}(T,S)\,S.$$

8. Find the Gaussian and mean curvatures of the plane, the sphere and the torus.

9. Prove that $H^2 \ge K$. When does equality hold?

10. Let $X, Y \in T_pM$ be orthonormal vectors. Prove that $H = \tfrac12\big(\mathrm{II}(X,X) + \mathrm{II}(Y,Y)\big)$.

11. Compute the coefficients $L_{ij}$ and the matrix $(L^j_i)$ for the helicoid $x(u, v) = (u\cos v,\ u\sin v,\ v)$.

12. Prove that if $X_1$ and $X_2$ are principal directions at a point with $\kappa_1 \ne \kappa_2$, then $\mathrm{II}(X_1, X_2) = 0$.

13. Prove that the mean curvature is the average of the normal curvatures: $H = \frac{1}{2\pi}\int_0^{2\pi}\mathrm{II}(X_\theta, X_\theta)\,d\theta$, where $X_\theta = \cos\theta\,X_1 + \sin\theta\,X_2$ for an orthonormal basis $X_1, X_2$ of $T_PM$.

14. Find the principal curvatures and principal directions of the saddle $x(u, v) = (u, v, uv)$ at the point $(u, v) = (1, 1)$.

15. Compute the Gaussian and mean curvature of the helicoid of Problem 11, and show that it is a minimal surface.

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Homework 9

Covers Lectures 22 to 24. Quiz 9 is based on these problems.

1. For a patch $x(r,s) = (r, s, f(r,s))$, prove that the area is given by $$\iint \sqrt{1 + f_r^2 + f_s^2}\,dr\,ds.$$

2. Using Problem 1, calculate the area of the hemisphere $x(r,s) = (r, s, \sqrt{1-r^2-s^2})$.

3. Prove that the $u^i$-curves are lines of curvature if and only if $g_{12} \equiv L_{12} \equiv 0$.

4. Prove that the circles of latitude and the meridians of a surface of revolution are lines of curvature.

5. Suppose $g_{12} \equiv 0$, and let $g = \det(g_{ij})$. Prove that $$K = -\frac{1}{2\sqrt{g}}\left(\frac{\partial}{\partial u^2}\Big(\frac{1}{\sqrt{g}}\frac{\partial g_{11}}{\partial u^2}\Big) + \frac{\partial}{\partial u^1}\Big(\frac{1}{\sqrt{g}}\frac{\partial g_{22}}{\partial u^1}\Big)\right).$$

6. Suppose $g_{12} \equiv 0$ and $g_{11} \equiv 1$. Prove that $$K\sqrt{g_{22}} + \frac{\partial^2 \sqrt{g_{22}}}{(\partial u^1)^2} = 0.$$

7. Determine which points on the patch $x(u,v) = (u, v, u^3 + v^3)$ have $K > 0$ and which have $K < 0$.

8. Prove that $n_1 \times n_2 = K\sqrt{g}\,n$, where $n_i = \partial n/\partial u^i$ and $g = \det(g_{ij})$.

9. What are the principal curvatures of a surface of revolution?

10. Let $\gamma(s)$ be a unit speed curve on a surface $M$. Its geodesic torsion is $\tau_g = \langle S, dn/ds \rangle$, where $S = n \times T$. Prove that $\tau_g \equiv 0$ if and only if $\gamma$ is a line of curvature.

11. Find the area of the torus $x(u, v) = \big((a + b\cos u)\cos v,\ (a + b\cos u)\sin v,\ b\sin u\big)$, where $0 < b < a$.

12. Compute the total curvature $\iint K\,dA$ of the part of the helicoid $x(u, v) = (u\cos v, u\sin v, v)$ with $-1 < u < 1$, $0 < v < 2\pi$.

13. Show that if $M = \mathbb{R}^2$ then $R^l_{ijk} = 0$ for all $i, j, k, l$, both intrinsically (from the metric, in polar coordinates) and extrinsically (from Gauss's equations).

14. Compute the Riemann curvature tensor $R^l_{ijk}$ of the cylinder $S^1 \times (0, 1)$, using the patch $x(\theta, t) = (\cos\theta, \sin\theta, t)$.

15. Show that the cylinder $x^2 + y^2 = 1$ is locally isometric to the plane, but that no open subset of the cylinder is isometric to an open subset of the unit sphere.

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