HW #7

Consider the parametric surface $\vec{x}(u,v)$ and the curve $\vec{\alpha}(t) = \vec{x}\bigl(u(t),v(t)\bigr)$ that lies on that surface. Without a loss of generality, suppose that $\vec{\alpha}$ is a unit-speed curve, that is $\vec{t} = \vec{\alpha}'$ is of unit length. Such a curve is called a geodesic if $\vec{t}\,'$ is perpendicular to the surface everywhere along the curve.

The definition of the geodesic imposes constraints on the functions $u$ and $v$ in the form of a pair of second order coupled ordinary differential equations called the geodesic equations. \begin{align*} &u'' + \Gamma_{11}^1 u'^2 + 2 \Gamma_{12}^1 u' v' + \Gamma_{22}^1 v'^2 = 0, \\ &v'' + \Gamma_{11}^2 u'^2 + 2 \Gamma_{12}^2 u' v' + \Gamma_{22}^2 v'^2 = 0. \end{align*} The following sequence of three mini-exercises show how the geodesic equations are derived.

  1. Show that \begin{align*} \vec{t} &= \vec{x}_u u' + \vec{x}_v v', \\ \vec{t}\,' &= (\vec{x}_{uu} u' + \vec{x}_{uv} v') u' + \vec{x}_u u'' + \cdots. \quad (\text{fill in the dots}) \end{align*}
  2. Recalling the definition of the Christoffel symbols, eliminate $\vec{x}_{uu}$, $\vec{x}_{uv}$ and $\vec{x}_{vv}$ from the above and arrive at \[ \vec{t}\,' = (\cdots) \vec{x}_u + (\cdots) \vec{x}_v + (\cdots) \vec{N}. \quad (\text{fill in the dots}) \]
  3. Explain how the geodesic differential equations (immediately) emerge from this..
  4. The figure below depicts a closed geodesic loop that winds around a torus four times. See if you can make a closed geodesic loop like that.
    A plotting trick: Although the geodesic lies exactly on the torus in the mathematical sense, the plotted geodesic won't be exactly on the torus due to numerical errors in solving the geodesic differential equations and graphical discretization. Without taking special measures, the geodesic will tend to appear somewhat “choppy” as parts of it sink below the torus's surface. In the current situation we can cheat a little to remedy that problem. Let's say the radii of the torus are $a$ and $b$. We do all the calculations pertaining to the geodesic with the correct $a$ and $b$. But when it comes to plotting the torus, we plot a slightly skinnier one with radii $a$ and $c$, where $c$ is something like 2% smaller than $b$. That way the geodesic stays entirely above the torus's surface. You can detect that if you look closely at the Figure above.
  5. Here is another closed geodesic loop. This one does not thread through the torus's hole.