We study holomorphic maps F from a smooth Levi non-degenerate real hypersurface \( M_{\ell }\subset {\mathbb {C}}^n \) into a hyperquadric \( {\mathbb {H}}_{\ell '}^N \) with signatures \( \ell \le (n-1)/2 \) and \( \ell '\le (N-1)/2,\) respectively. Assuming that \( N - n < n - 1,\) we prove that if \( \ell = \ell ',\) then F is either CR transversal to \( {\mathbb {H}}_{\ell }^N \) at every point of \( M_{\ell },\) or it maps a neighborhood of \( M_{\ell } \) in \( {\mathbb {C}}^n \) into \( {\mathbb {H}}_{\ell }^N.\) Furthermore, in the case where \( \ell ' > \ell ,\) we show that if F is not CR transversal at \(0\in M_\ell ,\) then it must be transversally flat. The latter is best possible.
