Math 2800: Differential Geometry
Mon, Wed 12:30 - 1:45pm -- Thackeray 703
Instructor.
Dr. Marta Lewicka (office hours in Thackeray 408, Mon 5:15pm - 6:00pm)
Grades.
Your final grade will be based on homeworks and two exams. Homework problems should be solved independently and the usual code of conduct applies.
Calendar.
29 Aug (Mon): First Class
5 Sept (Mon): Labor day - no classes
17 Oct (Mon): Fall Break - no classes
18 Oct (Tue): Monday schedule class and Midterm I
23 Nov (Wed): Thanksgiving break - no classes
7 Dec (Wed): Last Class
14 Dec (Wed): Midterm II
Homework
Homework 1 (due October 10)
1. Calculate the first and second fundamental forms of the following surfaces and determine their types:
(i) $\sigma = (\mbox{sinh}\,x\,\mbox{sinh}\,y, \mbox{sinh}\,x \,\mbox{cosh}\,y, \mbox{sinh}\, x)$,
(ii) $\sigma = (x-y, x+y, x^2+y^2)$,
(iii) $\sigma = (\mbox{cosh}\,x, \mbox{sinh}\,x, y)$,
(iv) $\sigma = (x, y, x^2+y^2)$.
2. Show that the following are equivalent conditions on a surface patch $S$ parametrised by $\sigma$, with the first fundamental form $I_S$:
$~~~$ (i) $\partial_2 E = \partial_1 G = 0$,
$~~~$ (ii) $\partial_{12}\sigma$ is parallel to $\vec N$,
$~~~$ (iii) The opposite sides of any quadrilateral on $S$ formed by curves (such curves are called the parameter curves) of the form $x\mapsto \sigma(x, y_0)$ and $y\mapsto \sigma(x_0, y)$ have the same lengths.
When these conditions are satisfied, the parameter curves as above are said to form a Chebyshev net. Show that then $S$ has a reparametrisation $\tilde \sigma(\tilde x, \tilde y)$ with the first fundamental form where $\tilde E = \tilde G = 1$ and $\tilde F = \cos\theta$, where $\theta(\tilde x, \tilde y)$ is a smooth function. Show that $\theta$ is the angle between the parameter curves of $\tilde\sigma$. Show that by putting $\hat u = \tilde u +\tilde v$, $\hat v=\tilde u - \tilde v$, the resulting reparametrisation $\hat\sigma$ has the first fundamental form given by: $\mbox{diag}(\cos^2(\theta/2), \sin^2(\theta/2))$.
3. Show that a surface diffeomorphic to an orientable surface is also orientable.
4. Let $t\mapsto \gamma(t)$ be a curve in $\mathbb{R}^3$, parametrised by arclength. Given a smooth map $t\mapsto\delta(t)\in\mathbb{S}^2$, we define the ruled surface $S$ by the parametrisation: $\sigma(x,y) = \gamma(x) + y\delta(x)$. Assume that $\delta=\gamma'$ and $\gamma''\neq 0$. Let $\epsilon>0$ be sufficiently small. Show that $S$ corresponding to the parameter range $y \in (\epsilon, 2\epsilon)$ is a regular surface and that it is locally isometric to a plane.
5. Let $\sigma: U\to\mathbb{R}^3$ be a regular parametrisation of a surface patch, such that $\vec N(0)$ is not parallel to the $(x,y)$ plane in $\mathbb{R}^3$. Show that near the point $p=\sigma(0)$, the surface $\sigma(U)$ is a part of the graph of some smooth function $\phi: U\to\mathbb{R}$.
6. Let $\Phi:U\to V$ be a diffeomorphism between open subsets of $\mathbb{R}^2$. Find an equivalent condition for $\Phi$ to be conformal.
7. Consider a spherical polygon on $\mathbb{S}^2$, that is a region formed by intersetion of $n\geq 3$ hemispheres. Compute the area of the polygon in terms of its interior angles.
8. Show that a map that is both conformal and equiareal must be a (local) isometry.
9. Show that the second fundamental form of a surface patch is unchanged by a reparametrisation of the patch which preserves its orientation. What is the effect on the second fundamental form of applying an isometry of $\mathbb{R}^3$? Or a dilation?
10. Let $S$ be a surface with parametrisation $\sigma: U\to\mathbb{R}^3$ and the corresponding first fundamental form $I_S$. Let $\gamma$ be a unit speed curve on $S$, whose base curve we denote by: $\tilde\gamma = (u(t), v(t))\in U$. Prove that the geodesic curvature of $\gamma$ can be computed by: $$ \kappa_g = (v'' u' - v' u'') \sqrt{\mbox{det} I_S} + A(u')^3 + B (u')^2v' + Cu' (v')^2 + D(v')^3,$$ where $A,B,C,D$ are functions of $E,F,G$ and their derivatives. Compute $A, B,C,D$ explicitly when $F=0$.
11. Compute the Christoffel symbols of the sphere $\mathbb{S}^2$ in the spherical coordinates parametrisation.
12. Let $S$ be the graph of a smooth function $f:U\to \mathbb{R}$. Compute the Gauss and mean curvatures $\kappa, H$ of $S$ in terms of $f$.
13. Show that the Weingarten map $\mathcal{W}$ of a surface $S$ satisfies the quatratic equation: $\mathcal{W}^2 - 2H\mathcal{W} + \kappa\mbox{Id} = 0$.
14. Let $S$ be an oriented surface. For all (sufficiently small) parameters $\lambda\in\mathbb{R}$ define: $S^\lambda = \{p+\lambda \vec N_p; ~ p\in S\}$.
$~~~$ (i) Show that $S^\lambda$ is a regular, oriented surface.
$~~~$ (ii) Find the formula for the unit normal vector to $S^\lambda$.
$~~~$ (iii) Find the principal curvatures and principal directions of $S^\lambda$ in terms of the same quantities for $S$.
$~~~$ (iv) Prove that the Gauss and mean curvatures of $S^\lambda$ are given by: $$\kappa^\lambda = \frac{\kappa}{1-2\lambda H + \lambda^2\kappa}, \qquad H^\lambda = \pm\frac{H-\lambda\kappa}{1-2\lambda H + \lambda^2\kappa}.$$
15. Assume that the first fundamental form of a surface parametrised by $\sigma$ has the form $E\mbox{Id}$. Show that the mean curvature $H=0$ if and only if $\Delta\sigma=0$.
Homework 2 (due November 9)
1. Describe four distinct geodesics on the hyperboloid $x^2 +y^2 - z^2=1$ passing through the point $(1,0,0)$.
2. A regular curve $\gamma$ with nowhere vanishing curvature on a surface $S$ is called a pregeodesic on S, if some reparametrization of $\gamma$ is a geodesic on $S$. Show that:
$~~~$ (i) $\gamma$ is a pregeodesic if an only if $\langle\gamma'', \vec N\times \gamma'\rangle =0$ along $\gamma$.
$~~~$ (ii) Any reperametrisation of a pregeodesic is a pregeodesic.
$~~~$ (iii) Any constant speed reparametrisation of a pregeodesic is a geodesic.
$~~~$ (iv) A pregeodesic is a geodesic if and only if it has constant speed.
3. Show that if $p$ and $q$ are two distinct points of the cylinder, then there are either exactly two or there are infinitely many geodesics (on the cylinder) with endpoints $p$ and $q$.
4. Show that the long great circle arc on the sphere $\mathbb{S}^2$, connecting the points $(1,0,0)$ and $(0,1,0)$, is not a local minimum of the length functional $\mathcal{L}$.
5. Show that there is no surface parametrization with the first and second fundamental forms given by: $I(x,y)=\mbox{diag}(1, \cos^2x)$ and $II(x,y)=\mbox{diag}(\cos^2x, 1)$.
6. Show that if a surface parametrization has first fundamental form given by $I(x,y) = e^{\lambda(x,y)} Id$, where $\lambda$ is a smooth function of $(x,y)\in B_\epsilon(0)\subset\mathbb{R}^2$, then its Gauss curvature $\kappa$ satisfies the equation: $\Delta \lambda + 2\kappa e^\lambda=0$.
7. Show that the Gauss curvature of the Mobius band equals $-1/4$ everywhere along its meridian circle.
8. Show that a compact surface whose Gauss curvature is positive everywhere, and whose mean curvature is constant, must be a sphere.
9. Show that the Euler characteristics of a compact surface is independent of its polygonal decomposition decomposition.
10. Let $\gamma$ be a positively-oriented unit-speed simple closed curve on a surface $S$ and let $v$ be a non-zero parallel vector field along $\gamma$. Prove that, going along $\gamma$ (once), the field $v$ rotates through an angle $2\pi - \int_{0}^{length(\gamma)}\kappa_g ~\mbox{d}s$.
Homework 3 (due December 16 in my mailbox)
1. Let $M$ and $N$ be two manifolds of dimensions $m$ and $n$, respectively. Define the manifold structure on the Cartesian product $M\times N$, find the transition maps, and prove such $M\times N$ is a manifold of dimension $m+n$.
2. We say that an atlas is a refinment of another atlas on $M$, if every chart of the first atlas is a restriction of some chart in the second atlas. Prove that any refinment defines the same smooth structure on $M$ (i.e. the identity map $id: (M,\mbox{first atlas})\to (M, \mbox{second atlas})$ is a diffeomorphism).
3. Given a $n$-dimensional manifold $M$, the total space of its tangent bundle is: $TM=\{(x,v); ~ x\in M, ~ v\in T_xM\}$. Introduce on $TM$ a structure of $2n$ dimensional smooth manifold. Further, prove that the total space of tangent bundle to the circle $M=\mathbb{S}^1$ is diffeomorphic to the cylinder $\mathbb{S}^1\times \mathbb{R}$.
4. Given a $n$-dimensional manifold $M$, the total space of its cotangent bundle is: $T^*M=\{(x,\omega); ~ x\in M, ~ \omega\in T_x^*M\}$. Introduce on $T^*M$ a structure of $2n$ dimensional smooth manifold.
5. A function $f\in\mathcal{C}^\infty (M)$ is called first integral of a vector field $v\in Vect(M)$ if $vf \equiv 0$. Prove that any first integral takes constant values on integral curves of the vector field. Prove that the vector field $v$ is tangent to the level surface of any first integral of this vector field.
6. Prove the following version of the commutator lemma from class, valid for nonautonomous ODEs. Two systems of ODEs on a manifold $M$: $$\frac{\partial x}{\partial t} = v(t,s,x), \qquad \frac{\partial x}{\partial s} = w(t,s,x) \quad \mbox{ for } x\in M,$$ admit (for sufficiently small $|t|$ and $|s|$) a unique common solution $x=x(t,s)$ with an arbitrary initial data $x(0,0) = x_0\in M$, if and only if the $(t,s)$-dependent vector fields $v$, $w$ satisfy: $$\frac{\partial v}{\partial s} - \frac{\partial w}{\partial t} = [v,w].$$
7. Let $M$ be a paracompact manifold. Show that there exists a partition of unity associated with some refinement of the atlas of $M$.
8. Let $T\in\mathcal{T}^p_q$ and let $1\leq k < l\leq p$. We define the operation of permutation of two upper indices by: $(\Pi^{kl} T)^{i_1\ldots i_k\ldots i_l\ldots i_p}_{j_1\ldots j_q} = T^{i_1\ldots i_l\ldots i_k\ldots i_p}_{j_1\ldots j_q}$. In a similar way, we define the operation $\Pi_{kl}$ of permuting two lower indices $1\leq k < l\leq q$ Prove that $\Pi^{kl}$ and $\Pi_{kl}$ are well defined linear operators from $\mathcal{T}^p_q$ to itself.
9. A pseudo-Riemannian metric on a $n$-dimensional manifold $M$ is a symmetric $(0,2)$ tensor $g\in\mathcal{T}^0_2$ such that $\det (g_{ij})_{1\leq i,j \leq n}\neq 0$. A manifold equipped with a pseudo-Riemannian metric is called a pseudo-Riemannian manifold. Show that, given a pair of positive integers $p,q$ satisfying $p+q=n$, the $n$-dimensional pseudo-Euclidean space $\mathbb{R}^{p,q}=\mathbb{R}^n$ becomes a pseudo-Riemannian manifold with the metric: $$ds^2 = (dx^1)^2+\ldots + (dx^p)^2 - (dx^{p+1})^2 -\ldots - (dx^n)^2.$$
10. Given a $n$-dimensional Riemann manifold $(M, g)$, show that the stationary points of the functional: $L(\gamma) = \int_{\gamma}|x'(t)|_g ~\mbox{d}t$, which associates to each arc-length parametrised curve $\gamma$ its length, are determined by the following system of differential equations: $$\frac{d^2x^k}{dt^2} + \sum_{i,j} \Gamma_{ij}^k(x) \dot x^i\dot x^j = 0 \qquad \forall k:1\ldots n \qquad \mbox{ where: } \qquad \Gamma_{ij}^k = \frac{1}{2}\sum_{s} g^{ks} \big(\frac{\partial g_{sj}}{\partial x^i} + \frac{\partial g_{is}}{\partial x^j} - \frac{\partial g_{ij}}{\partial x^s}\big),$$ and where we denote: $g^{ks} = (g^{-1})_{ks}$.
11. Let $T\in\mathcal{T}^p_q$ and for every $k=1\ldots n$ set: $$\nabla_k T^{i_1\ldots i_p}_{j_1\ldots j_q}:= \frac{\partial T^{i_1\ldots i_p}_{j_1\ldots j_q}}{\partial x^k} + \sum_s \Gamma_{ks}^{i_1} T^{s i_2\ldots i_p}_{j_1\ldots j_q} + \ldots + \sum_s \Gamma_{ks}^{i_p} T^{i_1\ldots i_{p-1} s}_{j_1\ldots j_q} - \sum_s \Gamma_{kj_1}^{s} T^{i_1\ldots i_p}_{s j_2\ldots j_q} - \ldots - \sum_s \Gamma_{kj_q}^{s} T^{i_1\ldots i_p}_{j_1\ldots j_{q-1} s}$$ (the formula that we derived in class). Show that writing $\nabla_v T = \sum_k v^k \nabla_{e^k} T$ where $\nabla_{e^k}T\in\mathcal{T}^p_q$ is given through: $(\nabla_{e^k} T)^{i_1\ldots i_p}_{j_1\ldots j_q} = \nabla_k T^{i_1\ldots i_p}_{j_1\ldots j_q}$, the properties of commuting with contractions and the Leibnitz rule are satisfied in the most general sense (stated in the theorem in class).
12. Define the operator $\nabla:\mathcal{T}^p_q\to \mathcal{T}^p_{q+1}$ by: $$(\nabla T)^{i_1\ldots i_p}_{j_1\ldots j_q k} = \nabla_k T^{i_1\ldots i_p}_{j_1\ldots j_q}.$$ Prove that $\nabla$ is well defined and linear.
13. Given two vector fields $v,w\in Vect(M)$, define the linear operator $R(v,w):Vect(M)\to Vect(M)$ by: $R(v,w) = [\nabla_v, \nabla_w] - \nabla_{[v,w]}.$ Prove that: $$R(v,w)u = -\sum_{l}\big(\sum_{i,j,k} R^l_{ijk} v^iw^ju^k \big) e^l.$$
14. Prove the following identities for contractions of Christoffel symbols with respect to the Levi-Civita connection: $$\sum_k \Gamma^k_{ik} = \frac{\partial\log \sqrt{\det g}}{\partial x^i}, \qquad \sum_{i,j} g^{ij}\Gamma^k_{ij} = -\frac{1}{\sqrt{\det g}}\sum_s \frac{\partial}{\partial x^s}\big(\sqrt{\det g} g^{sk}\big), \quad \forall k.$$
15. Prove that the covariant divergence $\sum_{i}\nabla_i v^i$ of a vector field $v\in Vect(M)$ with respect to the Levi-Civita connection, can be written in the following form: $$\sum_{i}\nabla_i v^i = \frac{1}{\sqrt{\det g}}\sum_i \frac{\partial}{\partial x^i} \big(\sqrt{\det g} v^i\big).$$