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This homework set is out of 40 points.
-
(15 points)
Find the Gauss, mean, and principle curvatures of
\[
T_3 :=\{ (x,y,y) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 + 3 = 4 \sqrt{x^2 + y^2} \}
\]
at the point \((1,0,0)\).
Hint:
This is the torus obtained by revolving the unit circle in the \(xz\)-plane centered at \((2,0,0)\)
about the \(z\)-axis.
It was featured in Homework 3.
-
(5+5+5 points)
Let \(0 < a < b\).
Find a parameterization \(\gamma(s)\) of the line segment connecting \((0,a)\) to \((0,b)\) which is unit speed with respect
to the Riemannian metric \(\frac{1}{y^2} dx^2 + \frac{1}{y^2} dy^2\).
Show that \(\gamma(s)\) satisfies the geodesic equations.
Show that every hyperbolic geodesic (in the sense of Homework 6) satisfies the geodesic equations.
Remark:
In this problem you are justifying the use of ``geodesic'' in the phrase ``hyperbolic geodesic''.
Here the geodesic equations generalize in the obvious way to Riemannian metrics:
if \(E\ du^2 + \ 2F\ du\ dv + G\ dv^2\) is a Riemannian metric, calculate the Christoffel symbols
as though \(E,F,G\) came from the first fundamental form of a surface.
Hint:
Your work in Homework 6 will be helpful here.
If \(T\) is an isometry of the hyperbolic plane and \(\gamma\) is a geodesic, so is \(T \circ \gamma\) (why?).
-
(5 points)
Show that \(T_4:= \{(s,t,u,v) \in \mathbb{R}^4 \mid s^2 + t^2 = 1 = u^2 + v^2\}\) is not
isometric to \(T_3\) from problem 1.
Hint:
Show that the plane is locally isometric to \(T_4\) but not to \(T_3\).
-
(5 points)
Suppose that \(S \subseteq \mathbb{R}^3\) is a smooth compact surface of positive genus.
Show that \(S\) has a point of negative Gauss
curvature.
Hint:
You may use Proposition 8.6.1.