  A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface.
1984. Journal of Differential Geometry, 19(1), 221-232.) says that a compact
minimal hypersurface $M^k$ of the sphere $S^{k+1}$ with $H^1(M)=0$, whose Gauss
map omits a neighborhood of an $S^{k-1}$ equator, is totally geodesic in
$S^{k+1}$. We develop a new proof strategy which can also obtain an analogous
result for codimension 2 compact minimal submanifolds of $S^{k+1}$.
