Normal connections induced by a framed hypersurface in the conformal space

We find two normal connections induced by the normal framing of a hypersurface $V_{n-1}$ in the conformal space $C_n$, and establish relationship between these connections and a Weyl connection which is also induced by the normal framing of $V_{n-1}$. We study two normal connections induced by a complete framing of a hypersurface $V_{n-1}$ in $C_n$. We establish relationship between geometries of a framed hypersurface $V_{n-1}$ of the conformal space $C_n$ and a quadratic hyperband of the projective space $\mathrm P_{n+1}$ associated with $V_{n-1}$.