Analytic saddle spheres in $$\mathbb {S}^3$$ are equatorial

A theorem by Almgren establishes that any minimal 2-sphere immersed in $$\mathbb {S}^3$$ S 3 is a totally geodesic equator. In this paper we give a purely geometric extension of Almgren’s result, by showing that any immersed, real analytic 2-sphere in $$\mathbb {S}^3$$ S 3 that is saddle, i.e., of non-positive extrinsic curvature, must be an equator of $$\mathbb {S}^3$$ S 3 . We remark that, contrary to Almgren’s theorem, no geometric PDE is imposed on the surface. The result is not true for $$C^{\infty }$$ C ∞ spheres.