  In this paper we show rigidity results for super-solutions to fully nonlinear
elliptic conformally invariant equations on subdomains of the standard
$n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We
emphasize that our results do not assume concavity assumption on the fully
nonlinear equations we will work with.
  This proves rigidity for compact connected locally conformally flat manifolds
$(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy
a fully nonlinear elliptic inequality and whose boundary is isometric to a
geodesic sphere $\partial D(r)$, where $D(r)$ denotes a geodesic ball of radius
$r\in (0,\pi/2]$ in $\mathbb S^n$, and totally umbilical with mean curvature
bounded below by the mean curvature of this geodesic sphere. Under the above
conditions, $(M,g)$ must be isometric to the closed geodesic ball
$\overline{D(r)}$.
  As a side product, in dimension $2$ our methods provide a new proof to
Toponogov's Theorem about the rigidity of compact surfaces carrying a shortest
simple geodesic. Roughly speaking, Toponogov's Theorem is equivalent to a
rigidity theorem for spherical caps in the Hyperbolic three-space $\mathbb
H^3$. In fact, we extend it to obtain rigidity for super-solutions to certain
Monge-Amp\`ere equations.
