Weak helix submanifolds of Euclidean spaces (Q1032515)

For a given submanifold \(M \subset \mathbb R^n\) of the Euclidean space a vector \(d \in \mathbb R\) is called a `helix direction vector' of \(M\) if the angle of \(d\) to all tangent spaces \(T_P M\) is constant. The manifold is called `weak \(r\)-helix' if there exists a set \(H(M)\) of \(r\) linearly independent helix direction vectors of \(M\). If this set \(H(M)\) can be extended to an \(r\)-dimensional subspace of \(\mathbb R^n\) the manifold \(M\) is called `strong \(r\)-helix'. The author gives an example that the two properties do not have to coincide for special \(2\)-manifolds \(M\) of \(\mathbb R^4\). He can show that all strong \(2\)-helices \(M \subset \mathbb R^4\) of dimension \(2\) can be generated by extensions of helices of \(\mathbb R^3\). The author also presents an example of a weak \(2\)-helix \(M\) which does not belong to the strong \(2\)-helices of above. This is done by discussing the solutions of special PDEs.