  We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere
with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary
conditions, the hypersurfaces are shown to be part of the equatorial spheres.
The lower bound $n(n-1)$ is critical in the sense that the hypersurface may
contain geodesic points and some natural differential operators are fully
degenerate at geodesic points. We overcome the difficulty by studying the
geometry of level sets of a height function, via new geometric inequalities.
Some rigidity results of hyperplanes and generalized cylinders are also
obtained for hypersurfaces with boundary and with nonnegative scalar curvature
in Euclidean space.
