Spherical minimal surfaces with minimal quadric representation in some hyperquadric (Q1346807)

By the second standard immersion of \(S^ 3\) into the Euclidean space \(SM(4)\) of 4-order real symmetric matrices, the authors give a characterization of the totally geodesic \(S^ 2\) and the Clifford torus in \(S^ 3\). Let \(x: M^ 2\to S^ 3\) be a compact, minimal surface of 3-dimensional unit sphere and \(f: S^ 3\to SM(4)\) the second standard immersion of \(S^ 3\), where \(SM(4)= \{P\in \text{gl} (4,\mathbb{R})\mid P^ t= P\}\). One then can combine both immersions, \(x\) and \(f\), to obtain an isometric immersion \(\Phi= f\circ x: M^ 2\to SM(4)\), called the quadric representation of \(x\) (or of \(M^ 2\)). The main result of this paper is as follows: The quadric representation \((M^ 2, \Phi)\) is minimal in some canonical hyperquadric of \(SM(4)\) if and only if \((M^ 2, x)\) is either totally geodesic or the Clifford torus in \(S^ 3\).