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This homework set is out of 40 points.

1. (10 points) Suppose that \(\sigma(u,v)\) is a surface patch defined on \(\mathbb{R}^2\) such that \(E\) depends only on \(u\), \(G\) depends only on \(v\), and \(F=0\). Show that there are functions \(u(x)\) and \(v(y)\) defined in \(\mathbb{R}\) such that \(\tau(x,y) = \sigma(u(x),v(y))\) is a local isometry.
   Remark: If we take \[ \sigma(u,v) = \big(\cos(v)\cos(u),\cos(v) \sin(u),2 \sin (v) \cos (\frac{u}{2}), 2 \sin (v) \sin (\frac{u}{2})\big) \] then it can be checked that \(E = 1\), \(F =0\), and \(G = 1 + 3 \cos^2 (v)\) and hence the range is locally isometric to \(\mathbb{R}^2\). This is a Klein bottle in \(\mathbb{R}^4\) similar to the one in problem 3 of homework set 4. This example is due to Charles Tompkins (Bull. AMS, 1941).
2. (10 points) Find all functions \(f:\mathbb{R} \to \mathbb{R}\) such that \(\sigma(u,v) = (f(u) \cos(v),f(u) \sin(v))\) is conformal.
3. (10 points) Let \(0 < a \leq 1\) be a constant. Calculate the first and second fundamental form of the surface patch \[ \sigma(r,\theta) := \big(a r \cos (\theta/a), a r \sin (\theta/a), r \sqrt{1-a^2} \big). \] defined for \((r,\theta) \in (0,\infty) \times \mathbb{R}\). Do these depend on \(a\)?
   Remark: If \(0 < a < 1\), the range of \(\sigma\) is a cone with the vertex removed; if \(a = 1\), it is \(\mathbb{R}^2\) with the origin removed.
4. (10 points) Let \(a\) and \(\sigma\) be as in problem 3. If \(\sigma(p) = \sigma(q)\), compute the matrix for the linear transformation \((D_q \sigma)^{-1} \circ D_p \sigma\) (with respect to the standard basis for \(\mathbb{R}^2_p\) and \(\mathbb{R}^2_q\)).