  A theorem by Almgren establishes that any minimal $2$-sphere immersed in
$\mathbb{S}^3$ 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 $\mathbb{S}^3$ that is saddle, i.e., of non-positive
extrinsic curvature, must be an equator of $\mathbb{S}^3$. We remark that,
contrary to Almgren's theorem, no geometric PDE is imposed on the surface. The
result is not true for $C^{\infty}$ spheres.
