Generalized helical immersions

Abstract

In thispaper, we assume thatallgeodesies are parametrized by thearclength. Let/ be an isometric immersion of a Riemannian manifold M into a Riemannian manifold M. If geodesies in M are viewed as specificcurvesin M, what are the shape of /(M)? Several geometricians studied this problem. K. Sakamoto characterized an isometric immersion / of a complete connected Riemannian manifold M into a Euclidean space or a sphere such thatevery in M is viewed as a helix in the ambient space and that the order and the Frenet curvatures of the helix are independent of the choice of the (cf.[15], [16]).In [5],D. Ferus and S. Schirrmacher investigated an isometric immersion/ of a compact connected Riemannian manifold M into a Euclidean space Rm satisfyingthe following condition: (A) Almost every in M is viewed as a generic helixin Rm. Here almost every geodesic means thatthe tangent vectorsof such geodesies fill the unit tangent bundle of M up to a closed setof measure zero and a generic helix means a helix of even order such that the closure of theimage coincides with the lowest dimensional Clifford torus containing it.In [4] and [5],they showed that the condition (A) is equivalent to the following two conditions, respectively: (B) / is extrinsicsymmetric in the sense of [4]. (C) The second fundamental form of / is parallel. In thispaper, we consider an isometric immersion / of a Riemannian manifold M into a Riemannian manifold M such thatevery geodesicin M is viewed as a helix in M, where theorder ofthe helixmay depend on the choice of the geodesic.We callsuch a immersion a generalized helicalimmersion and the highest order of those helicesthe order off. First,we show that allisometric immersions with parallelsecond fundamental form are generalized helical.Conversely, itis very interesting to investigatein what case a generalized helicalimmersion has the