An extrinsic decomposition theorem and a slant tube theorem for a curvature netted hypersurface (Q5943573)

Let \(f:M\to N^{n+1}(c)\) be a connected, complete hypersurface in a space of constant curvature \(c\), \(\mu(p)\) the number of distinct principal curvatures of \(f\) at the point \(p\in M\), \(g:=\max_{p\in M} \mu(p)\), \(W_g\) the open subset \(\{p\in M\mid\mu (p)=g\}\), and \({\mathfrak F}_1, \dots,{\mathfrak F}_g\) the ``principal foliations'' on \(W_g\) associated to the principal curvatures \(\lambda_1, \dots, \lambda_g\) of \(f\). Many authors have studied Dupin hypersurfaces, i.e. the case \(W_g=M\) such that each \(\lambda_i\) is constant along the leaves of \({\mathfrak F}_i\). For instance, in this case \(g\) can only take the values \(1,2,3,4\) or 6. For \(c\geq 0\) and \(\dim{\mathfrak F}_i\geq 3\) the author is concerned with the following generalization: He only assumes that \(W_g\) is a dense subset, but that the ``curvature net'' \(({\mathfrak F}_1, \dots,{\mathfrak F}_g)\) can be extended to the entire manifold \(M\). Then there can be constructed examples with arbitrary \(g\in\mathbb{N}\). Under some further technical assumptions he shows that \(M\) is covered by a ``mixed warped product'' of spheres (corresponding to the lift of the curvature net). Furthermore, in the case \(c=0\) he defines a ``slant focal map'' \(f_i:M/{\mathfrak F}_i\to \mathbb{R}^{n+1}\) for each \(i\) and shows that the hypersurface \(f\) is a ``slant tube'' with a variable radius function about \(f_i\).