Equal area

[Tsakanikas Nickos]

1. Let \( \displaystyle c : I \subset \mathbb{R} \longrightarrow \mathbb{R}^{3} \) be a closed curve parametrized by arc length and with curvature \( \displaystyle \kappa(s)>0 \, , \, \forall s \in I \). If the curve \( \displaystyle \gamma : I \subset \mathbb{R} \longrightarrow S^{2} \, , \, \gamma (s) = \vec{n} (s) \) is simple and defines a simple region on \( \displaystyle S^{2} \), then it divides \( \displaystyle S^{2} \) in two regions with equal areas.
2. Let \( \displaystyle S \subset \mathbb{R}^{3} \) be a regular surface homeomorphic to a sphere, with positive Gaussian curvature. Let \( \displaystyle \Gamma \subset S \) be a simple closed geodesic in \( \displaystyle S \), and let \( \displaystyle A \) and \( \displaystyle B \) be the two regions of \( \displaystyle S \) which have \( \displaystyle \Gamma \) as a common boundary. Let \( \displaystyle N : S \longrightarrow S^2 \) be the Gauss map of \( \displaystyle S \) and suppose that \( \displaystyle N \) is 1-1. Prove that \( \displaystyle N(A) \) and \( \displaystyle N(B) \) have the same area.