An intrinsic geometry of second type of surfaces \(X_ 2\subset E_ 4\) (Q1357941)

On a non-developable surface \(X_2\) in Euclidean space \(E_4\) in addition to the usual Levi-Civita connection \(\nabla\) and normal connection \(\nabla^\perp\) a tangent connection of second type is considered, which is the Levi-Civita connection of the nondegenerate symmetric tensor \(\gamma_{ij}= g_{\alpha\beta} g^{kl}h^\alpha_{ik} h^\beta_{jl}\); here \(h\) is the second fundamental tensor. It is established that the geometry of second type on \(X_2\) (i.e., with metric tensor \(\gamma\)) is the geometry of a space of constant curvature and \(X_2\) is determined, up to the position in \(E_4\), by two functions on this space (distances of tangent hyperplanes from the origin). As a generalization it is announced that similarly the geometry of second type of a submanifold \(X_m\) in Euclidean space \(E_n\) is in general the geometry of an \(m\)-dimensional locally symmetric space.