Differential Geometry Homework assignment 2 (due 2/11/20)

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This homework set is out of 30 points. Each challenge problem may be used to replace a regular problem and offers the possibility of 5 points of extra credit. (You may only submit solutions to 3 problems.)

  1. (4 points) Compute the curvature and torsion of \(\gamma(t) = (a \cos(t),a\sin (t), bt)\). (3 points) Give a general formula for a unit speed parameterized curve with constant nonzero curvature \(\kappa\) and constant torsion \(\tau\). Your answer should be in the form \(T \circ (x(s),y(s),z(s))\) where \(x(s)\), \(y(s)\), and \(z(s)\) are functions specified in terms of \(\kappa\) and \(\tau\) and \(T\) is an arbitrary isometry of \(\mathbb{R}^3\). (3 points) Find a parameterization of the curve \(\alpha\) with curvature \(1\), torsion \(-1\), with \(\gamma(0) = (1/2,0,0)\), and whose tangent vector at \(\alpha(0)\) is \(\mathbf{k}\).

    Solution We have \[ \dot \gamma = (-a \sin (t), a \cos(t),b) \qquad \ddot \gamma = (-a \cos(t),-a \sin(t),0)\] \[ \dot \gamma \times \ddot \gamma = (ab \sin (t), - ab \cos(t),a^2) \qquad \dddot \gamma = (a \sin (t), - a \cos(t),0) \] Which gives \[ \kappa = \frac{|a|}{a^2 + b^2} \qquad \tau = \frac{b}{a^2 + b^2}. \] Given any \(\kappa > 0\) and \(\tau\), we can solve to obtain \[ a = \kappa (1 + \frac{\tau^2}{\kappa^2}) \qquad b = \tau (1 + \frac{\tau^2}{\kappa^2}) \] Since curves in \(\mathbb{R}^3\) are uniquely determined by their curvature and torsion functions, it follows that any curve with constant curvature \(\kappa > 0\) and torsion \(\tau\) is of the form \(T \circ (a \cos(t),a \sin(t),bt)\) where \(T\) is an isometry of \(\mathbb{R}^3\). For the last part, if \(\kappa = 1\) and \(\tau = -1\), we have \(a = 2\) and \(b = -2\). If we let \(\gamma(t) = (2 \cos (t),2 \sin (t),-2t)\), then \(\dot \gamma(0) = (0,2, -2)\). This gives the unit tangent vector \((0,\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})\). If we set \[ T(x,y,z) = \big(x-\frac{3}{2},\frac{\sqrt{2}(y + z)}{2},\frac{\sqrt{2}(y-z)}{2}\big) \] then \(T\) is an isometry and \[ \alpha (t) = T(\gamma(t)) = \big( 2 \cos(t) - \frac{3}{2}, \sqrt{2}(\sin (t) - t), \sqrt{2} (\sin(t) + t) \big) \] is the desired curve.
  2. (10 points) A plane curve \(\gamma:(0,\infty) \to \mathbb{R}^2\) with unit speed has signed curvature given by \(\kappa_s (s) = \frac{1}{2 \sqrt{s}}\). If \(\gamma(\pi^2) = (0,0)\) and \(\gamma'(\pi^2) = (1,0)\), what is \(\gamma(4\pi^2)\)?

    Solution The turning angle \(\phi\) satisfies \(\phi(s) - \phi(\pi^2) = \int_{\pi^2}^s \frac{1}{2 \sqrt{t}}\ dt = \sqrt{s} - \pi\). Setting \(\phi(\pi^2) = 0\) is consistent with out initial conditions \((\cos(\phi(\pi^2)),\sin(\phi(\pi^2))) = (1,0)\). This gives \(\phi = \sqrt{s} - \pi\). Now \[ \gamma(4 \pi^2) = \big( \int_{\pi^2}^{4 \pi^2} \cos(\sqrt{s} - \pi)\ ds, \int_{\pi^2}^{4 \pi^2} \sin(\sqrt{s} - \pi)\ ds \big) \] These integrals can be evaluated using the substitution \(x = \sqrt{s} - \pi\), noting that \(s = (x+\pi)^2\) and \(ds = 2(x + \pi)\ dx\). This gives, e.g. \[ \int_{\pi^2}^{4 \pi^2} \cos( \sqrt{s} - \pi)\ ds = \int_0^\pi 2 (x+\pi) \cos(x)\ dx = 2(x+\pi) \sin (x) \big|_{0}^\pi - 2\int_0^\pi \sin (x)\ dx = -4 \] A similar computation gives \(\int_{\pi^2}^{4\pi^2} \sin ( \sqrt{s} - \pi) \ ds = 6 \pi\), yielding \(\gamma(4 \pi^2) = (-4,6 \pi)\).
  3. (5 points) Show that if \(T:\mathbb{R}^n \to \mathbb{R}^n\) is a linear transformation, if \(\alpha:(c,d) \to \mathbb{R}^n\) is differentiable at \(t_0 \in (c,d)\), and if \(\beta (t) = T(\alpha(t))\), then \(\beta'(t_0) = T(\alpha'(t_0))\). (5 points) Show that if \(n=3\), if \(T\) is in \(SO(3)\), and if \(\alpha\) is unit speed and has nowhere vanishing curvature, then the curvature and torsion functions for \(\alpha\) and \(\beta\) are the same. Remark: \(SO(3)\) is the set of all linear transformations \(T\) of \(\mathbb{R}^3\) which satisfy \(T(\mathbf{a}) \cdot T(\mathbf{b}) = \mathbf{a} \cdot \mathbf{b}\) and \(T(\mathbf{a} \times \mathbf{b}) = T(\mathbf{a}) \times T(\mathbf{b})\).

    Solution If \((a_{i,j})_{i,j=1}^n\) is matrix for \(T\) and, e.g., \(\alpha_i\) is the \(i\)th coordinate of \(\alpha_i\), then \(\beta_i(t) = T(\alpha(t)) = \sum_{j=1}^n a_{i,j} \alpha_j(t)\). Differentiating we get \(\beta'_i(t) = \sum_{j=1}^n a_{i,j} \alpha_j'(t)\) and thus \(\beta'(t) = T(\alpha'(t))\). To see the second part, we have that \(\dot \beta = T \circ \dot \alpha\), \(\ddot \beta = T \circ \ddot \alpha\), and \(\dddot \beta = T \circ \dddot \alpha\). Since \(T\) is in \(SO(3)\), we have that for any \(\mathbf{v}\), \(\|T(\mathbf{v})\| = \|\mathbf{v}\|\). In particular, \(\|\dot \alpha\| = \|T \circ \dot \alpha\| = \|\dot \beta\|\). Furthermore \[ \|\dot \beta \times \ddot \beta = (T \circ \dot \alpha) \times (T \times \ddot \alpha) = T \circ (\dot \alpha \times \ddot \alpha) \] \[ (\dot \beta \times \ddot \beta)\cdot \dddot \beta = (T \circ (\dot \alpha \times \ddot \alpha)) \cdot (T \circ \dddot \alpha) = (\dot \alpha \times \ddot \alpha) \cdot \dddot \alpha \] In particular \(\|\dot \beta \times \ddot \beta\| = \|\dot \alpha \times \ddot \alpha\|\). Hence the curvature and torsion functions for \(\alpha\) and \(\beta\) coincide: \[ \frac{\|\dot \beta \times \ddot \beta\|}{\|\dot \beta\|^3} = \frac{\|\dot \alpha \times \ddot \alpha\|}{\|\dot \alpha\|^3} \qquad \frac{(\dot \beta \times \ddot \beta) \cdot \dddot \beta}{\|\dot \beta \times \ddot \beta\|^2} = \frac{(\dot \alpha \times \ddot \alpha) \cdot \dddot \alpha}{\|\dot \alpha \times \ddot \alpha\|^2} \]

Challenge problems The goal with these four problems is to prove the Jordan Curve Theorem and Hopf's Umlaufsatz by first proving "pixelated" versions of these theorems and then using them to establish the main results. Unlike their smooth analogs, the pixelated forms of these theorems allow inductive proofs.

For the purpose of these exercises, a planar lattice curve is a function \(\gamma:[a,b] \to \mathbb{R}^2\) such that:

  1. (15 points) Show that if \(\gamma\) is a planar lattice curve which is simple and closed, then \(\gamma\) satisfies the Jordan curve theorem: \(\mathbb{R}^2 \setminus \mathrm{range}(\gamma)\) is a disjoint union of two path connected sets, one which is bounded and the other which is unbounded. Recall that a subset \(A\) of \(\mathbb{R}^n\) is path connected if whenever \(x_0,x_1 \in A\), there is continuous function \(\alpha:[0,1] \to A\) such that \(\alpha(0) = x_0\) and \(\alpha(1) = x_1\).
  2. (15 points) Suppose that \(\gamma:[a,b] \to \mathbb{R}^2\) is a planar lattice curve which is simple and closed.
    1. Show that there is a function \(\phi:[a,b] \cap \mathbb{Z} \to \mathbb{R}\) such that:
      • if \(a \leq i < b\) is an integer and \(i < t < i+1\), then \(\dot \gamma (t) = (\cos \phi(i) , \sin \phi(i))\);
      • \(\phi(b) - \phi(a)\) is an integer multiple of \(2 \pi\);
      • if \(a \leq i < b\), then \(|\phi(i) - \phi(i+1)| \leq \pi/2\).
    2. Show that if \(\phi\) is as above, then \(\phi(b) - \phi(a) = \pm 2 \pi\).
  3. (15 points) Use the first challenge problem to prove the Jordan curve theorem for all regular simple closed curves. Remark: this problem requires some knowledge of analysis (e.g. MATH 4130/4140).
  4. (15 points) Use the second challenge problem to prove Hopf's Umlaufsatz. Remark: this problem requires some knowledge of analysis (e.g. MATH 4130/4140).