HW #6

The following familiar definitions and formulas are collected here for convenient reference. Here $x(u,v)$ represents a parameterized surface which is as many times differentiable as needed. We have \[ e = \vec{x}_{uu} \cdot \vec{N}, \quad f = \vec{x}_{uv} \cdot \vec{N}, \quad g = \vec{x}_{vv} \cdot \vec{N}, \] where $\vec{N}$ is the unit normal to the surface. Furthermore, we have \begin{align*} \vec{x}_{uu} &= \Gamma^1_{11} \vec{x}_u + \Gamma^2_{11} \vec{x}_v + e \vec{N} \\ \vec{x}_{uv} &= \Gamma^1_{12} \vec{x}_u + \Gamma^2_{12} \vec{x}_v + f \vec{N} \\ \vec{x}_{vv} &= \Gamma^1_{22} \vec{x}_u + \Gamma^2_{22} \vec{x}_v + g \vec{N}, \end{align*} where $\Gamma^k_{ij}$ are the Christoffel symbols.
  1. Show that $(\vec{x}_{uu})_v \cdot \vec{N} = f \, \Gamma^1_{11} + g \, \Gamma^2_{11} + e_v$.
  2. Show that $(\vec{x}_{uv})_u \cdot \vec{N} = e \, \Gamma^1_{12} + f \, \Gamma^2_{12} + f_u$.
  3. Redo the above with $(\vec{x}_{vv})_u \cdot \vec{N}$ and $(\vec{x}_{uv})_v \cdot \vec{N}$.
  4. Conclude that \begin{align} e_v - f_u &= e \, \Gamma^1_{12} - f \, (\Gamma^1_{11} - \Gamma^2_{12}) - g \, \Gamma^2_{11}, \\ f_v - g_u &= e \, \Gamma^1_{22} - f \, (\Gamma^1_{12} - \Gamma^2_{22}) - g \, \Gamma^2_{12}. \end{align}

    Remark 1: These are called the Mainardi–Codazzi equations. Considering that the Christoffel symbols are determined entirely by the metric coefficients $E$, $F$, $G$, these equations link the functions $E$, $F$, $G$ to $e$, $f$, $g$. A special case where $F=0$ appears on page 127 of our textbook.

    Remark 2: Our textbook refers to these as the Codazzi–Mainardi equations which is a bit unfair since Mainardi published these equations 12 years before Codazzi. Some authors call these the Peterson–Mainardi–Codazzi Equations because Peterson published these in his thesis 12 years before Mainardi! See the original publication dates in the references section in Math World.

  5. Plot the parametric surfaces \[ H(u,v) = \begin{pmatrix} \sinh v \sin u \\ - \sinh v \cos u \\ u \end{pmatrix}, \qquad C(u,v) = \begin{pmatrix} \cosh v \cos u \\ \cosh v \sin u \\ v \end{pmatrix}, \] over the range $-\pi < u < \pi$ and $-1 < v < 1$ and note that $H$ is a helicoid and $C$ is a catenoid.

    Then consider the blended surface $x(t) = H \cos t + C \sin t$, where the blending parameter $t$ varies between $0$ and $\pi/2$. Note that $x(0)= H$ and $x(\frac{\pi}{2})=C$. The textbook (page 168) calls this is a helcat (helicoid + catenoid).

    Show that the the metric coefficients $E$, $F$, $G$ of the helcat are independent of $t$. Thus, as $t$ varies, the helcat bends but does not stretch. Produce an animation that shows how the helix morphs into the catenoid. A few stages of this morphing are shown in Figure 4.4 on page 169 of the textbook.