Evolute surfaces in \(E^ 4\). (Q1810027)

Let \(M\) be a smooth 2-surface in Euclidean space \(E^{4}.\) The surface \(\overline{M}\) enveloping the family of normal 2-planes to \(M\) is the evolute of \(M.\) The tangent 2-planes at a point \(p \in M\) and at the corresponding point \(f(p)\in \overline{M}\) are mutually orthogonal, and the vector \(\overrightarrow{p f(p)} = \tau(p),\) \(\tau(p) \in T_{p}^{\bot}M,\) is the normal vector to \(M.\) The following results are proved: Theorem 1. If the evolute \(\overline{M}\) for a smooth 2-surface \(M \subset E^{4}\) exists, then the normal connection of the surface \(M\) is locally flat. Theorem 2. If a 2-surface \(\overline{M} \subset E^{4}\) is the evolute for some 2-surface \(M,\) then the Levi-Civita connection of \(\overline{M}\) is locally flat. Theorem 3. Minimal surfaces have no evolutes.