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 Last modified: 12:34 PM - 2 December, 2021

1. Let \(f : S_1 \to S_2\) denote a smooth function that between surfaces that is 1-1, onto, its inverse is smooth, and so that \(f^*\langle \mathbf{v}, \mathbf{w} \rangle = \langle \mathbf{v}, \mathbf{w} \rangle\) (called a local isometry) and let \(\sigma_1 : U \to S_1\) denote a surface patch on \(S_1\). Let \(\sigma_2 = f \circ \sigma_1\)

   1. Show that \((\sigma_2)_x = D_p(f) (\sigma_1)_x\) and \((\sigma_2)_y = D_p(f) (\sigma_1)_y\)

   2. Show that if \((\sigma_1)_x \times (\sigma_1)_y\neq 0\), then \((\sigma_2)_x \times (\sigma_2)_y\neq 0\)

   3. We can then treat \(\sigma_2\) as a surface patch for \(S_2\). Show that if \(E_1\), \(F_1\), \(G_1\) denote the entries of the matrix of the first fundamental form with respect to \(\sigma_1\) and \(E_2\), \(F_2\), \(G_2\) denote the entries of the matrix of the first fundamental form with respect to \(\sigma_2\), then \(E_1= E_2\), \(F_1 = F_2\), and \(G_1 = G_3\).

   4. Why does \(f\) map geodesics to geodesics?

2. Prove that the geodesic curvature of a curve in a plane (treated as a surface in \(\mathbb{R}^3\)) is equal to the plane curvature.

3. Compute the normal curvature of any curve on the sphere at a point in the region covered by the surface patch, \(\sigma(x,y) = (x, y, \sqrt{1 - x^2 - y^2})\). Can you interpret the answer physically? Using this, prove that curves on the sphere that have constant geodesic curvature are circles.