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

A real hypersurface in the complex quadric $Q^m=SO_{m+2}/SO_mSO_2$ 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 $\mathfrak A$-principal Hopf hypersurface in $Q^m$, $m\geq3$ 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 $Q^m$, $m\geq3$, is also obtained.