Disproving an old A. D. Alexandrov's hypothesis on a characterization of the 2-sphere, Y. Martinez-Maure constructed a curious object in Euclidean 3-space $R^3$: a closed oriented surface M that has the following properties:
M is a smooth saddle surface at each of its points (except for four points called ‘horns’): every point x of M distinct from a horn is a hyperbolic point of M (the Gaussian curvature of M is negative at x).
M possesses (as smooth strictly convex surfaces) a well-defined smooth support function on $S^2$. In particular, its Gauss map $M → S^2$ is injective.
See :
M G Knyazeva and Gayane Yu Panina, Russian Mathematical Surveys Volume 63, 2008.
and/or
Y. Martinez-Maure, Contre-exemple à une caractérisation conjecturée de la sphère, Comptes Rendus de l'Académie des Sciences de Paris 332, Série I, 2001, 41-44.