$\mathfrak{A}$-principal Hopf hypersurfaces in complex quadrics

A real hypersurface in the complex quadric $Q^m=S{O}_{m+2}\mathrm{/}S{O}_{m}S{O}_2$ is said to be $\mathfrak{A}$-principal if its unit normal vector field is singular of type $\mathfrak{A}$-principal everywhere. In this paper, we show that a $\mathfrak A$-principalHopfhypersurfacein$Q^m$,$m\geq3$ is an open part of a tube around a totally geodesic $Q^{m+1}$ in $Q^m$. We also show that such real hypersurfaces are the only contact real hypersurfaces in $Q^m$. %, this answers affirmatively a question posted by Berndt (cf. \cite{berndt1})}. The classification for pseudo-Einstein real hypersurfaces in $Q^m$, $m\ge 3$, is also obtained.