91Ó°ÊÓ

Show that a spherical triangle (with side-lengths less than \(\pi\) ) must be contained in some open hemisphere of \(S^{2}\).

For every spherical triangle \(\Delta=A B C\), show that \(a

Show that any Möbius transformation \(T\) on \(\mathbf{C}_{\infty}\) which is not the identity has one or two fixed points. Show that the Möbius transformation corresponding (under the stereographic projection map) to a rotation of \(S^{2}\) through a non-zero angle has exactly two fixed points \(z_{1}\) and \(z_{2}\), where \(z_{2}=-1 / \bar{z}_{1}\). If now \(T\) is a Möbius transformation with two fixed points \(z_{1}\) and \(z_{2}\) satisfying \(z_{2}=-1 / \bar{z}_{1}\), prove that either \(T\) corresponds to a rotation of \(S^{2}\), or one of the fixed points, say \(z_{1}\), is an attractive fixed point, i.e. for \(z \neq z_{2}\), we have that \(T^{n} z \rightarrow z_{1}\) as \(n \rightarrow \infty\).