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This homework set is out of 40 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 4 problems.)

1. (10 points) Show that \(\{(x,y,z) \in \mathbb{R}^3 \mid \sin (x) + \sin (y) = \sin (z)\}\) is a smooth surface. Remark: a portion of this surface is illustrated on the course webpage.
   
   Solution Define \(f(x,y,z) = \sin(x) + \sin(y) - \sin(z)\). If \(\nabla f = (\cos(x),\cos(y),-\cos(z))\) is \((0,0,0)\), then \(|\sin(x)| = |\sin(y)| = |\sin(z)| = 1\). It follows that \(\sin(x) + \sin(y)\) is even while \(\sin(z)\) is odd. Thus \(0\) is a regular value of \(f\) and consequently \(f^{-1}(0)\) is a smooth surface.
2. (10 points) Show directly that every point on \(\{(s,t,u,v) \in \mathbb{R}^4 \mid s^2 + t^2 = u^2 + v^2 = 1\}\) has a neighborhood homeomorphic to \((a,b) \times (c,d)\) for some \(a < b\) and \(c < d\). Remark: This is a torus. Unlike the familiar tori in \(\mathbb{R}^3\), this one is "flat". (We will return to this remark when we are in a position to make sense out of it.)
   
   Solution Define \(\sigma : \mathbb{R}^2 \to \mathbb{R}^4\) by \[ \sigma(\theta,\phi) = (\cos(\theta),\sin(\theta), \cos(\phi),\sin (\phi)) \] If \((s,t,u,v) \in \mathbb{R}^4\) are such that \(s^2 + t^2 = u^2 + v^2 = 1\), then there exist \((\theta_0,\phi_0) \in \mathbb{R}^2\) such that \(\sigma(\theta_0,\phi_0) = (s,t,u,v)\). Since \(\sigma\) is a diffeomorphism when restricted to \((\theta_0-\pi,\theta_0 + \pi) \times (\phi_0-\pi,\phi_0+\pi)\), we are done.
3. (10 points) Define a continuous bijection between the following two sets: \[\{(s,t,u,v) \in \mathbb{R}^4 \mid s^2 + t^2 = u^2 + v^2 = 1\}\] \[\{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 + 3 = 4 \sqrt{x^2 + y^2}\}\] Hint: The latter set is a standard torus in \(\mathbb{R}^3\) which is symmetric about the \(z\)-axis.
   
   Solution Let \(S \subseteq \mathbb{R}^4\) and \(T \subseteq \mathbb{R}^3\) denote these surfaces. Define \(f:S \to \mathbb{R}^3\) and \(g:T \to \mathbb{R}^4\) by \[ f(s,t,u,v) = ((2+s)u,(2+s)v,t) \] \[ g(x,y,z) = (\frac{x^2 + y^2 + z^2 - 5}{4},z,\frac{4x}{x^2 + y^2 + z^2 + 3},\frac{4y}{x^2 + y^2 + z^2 + 3}) \] Clearly these functions are continuous. It it also readily checked that \(g\) is the inverse of \(f\) and hence \(f\) is a bijection from the first surface to the second.
4. (6+2+2 points) Suppose that \(\gamma : [a,b] \to \mathbb{R}^2\) is a regular simple closed plane curve parameterized by arc length. Show that if \(\kappa_s(s) > 0\) for all \(s \in [a,b]\), then \(\dot \gamma :[a,b) \to \mathbb{R}^2\) is one-to-one. Give examples which show that this statement becomes false if either the hypotheses "simple" or "closed" are omitted.
   
   Solution Since \(\kappa_s > 0\) and since \(\phi' = \kappa_s\), \(\phi(s)\) is increasing. Since \(\gamma\) is simple and closed, Hopf's Umlaufsatz implies that \(\phi(b) - \phi(a) = 2 \pi\). Therefore \(\dot \gamma(s) = (\cos(\phi(s)),\sin(\phi(s)))\) is one-to-one on \([a,b)\). For the counterexamples: take reparameterizations of \(\alpha(t) = (e^t \cos(t),e^t \sin(t))\) for \(0 \leq t \leq 6 \pi\) and \(\beta(t) = ((1+2 \cos(t)) \cos(t),(1+2\cos(t)) \sin(t))\). The former curve is a portion of a spiral and the later is a limaçon. It is readily checked that \(\kappa_s > 0\) for these curves and that \(\dot \alpha\) and \(\dot \beta\) are not one-to-one.

Challenge problems

1. (7+8 points) Suppose that \(\gamma :[0,T] \to \mathbb{R}^2\) is a regular closed plane curve parameterized by arclength. Show that if \(\gamma\) has only finitely many points of self intersection, then there are piecewise regular simple closed curves \(\alpha_i : [0,T_i] \to \mathbb{R}^2\) for \(i = 1, \ldots,k\) such that:
   • each point on \(\gamma\) is on some \(\alpha_i\) and each point on an \(\alpha_i\) is on \(\gamma\);
   • if \(\gamma(s)\) is not a self intersection point of \(\gamma\), then there is a unique \(i\) and \(t \in [0,T_i)\) such that \(\alpha_i(t) = \gamma(s)\) and \(\dot \alpha_i(t) = \dot \gamma(s)\).
   Show that \(\int_0^T \kappa_s \ ds = 2 \pi (p-n)\) where \(p\) is the number of \(\alpha_i\) which are positively oriented and \(n\) is the number of \(\alpha_i\) which are negatively oriented.
2. (15 points) Suppose that \(\gamma : [0,T] \to \mathbb{R}^2\) is a unit speed simple closed curve. Show that if \(\kappa_s(s) \geq 0\) for all \(s \in [0,T]\), then \(\gamma\) is convex.