Flat approximations of hypersurfaces along curves

Abstract

Given a smooth curve γ in some m-dimensional surface M in R m+1 , we study existence and uniqueness of a flat surface H having the same field of normal vectors as M along γ, which we call a flat approximation of M along γ. In particular, the well-known characterisation of flat surfaces as torses (ruled surfaces with tangent plane stable along the rulings) allows us to give an explicit parametric construction of such approximation.

The geometric Cauchy problem for rank-one submanifolds

“…In particular, in a joint work with Irina Markina [12], the author has studied the case of developable hypersurfaces in R m+1 (the case m = 2 is well-known, see e.g. [8, p. 195-197]).…”

“…The purpose of this note is twofold. On one hand, we aim to give a new and simpler proof of the main theorem in [12]. At the same time, we intend to generalize such result to the whole class of developable submanifolds of R m+n .…”

Generating Generalized Cylinder with Geodesic Base Curve According to Darboux Frame

“…Approximating by an osculating cylinder (22) has nice geometrical properties: globally free of singularities, achieved along a geodesic path, some geometric quantities on both cylinder and the surface along the geodesic curve are congruent up to orientation (2.2), locally like approximating by the ribbon [28], and the cylinder is easy to modeling and reasoning from a mathematical or manufacturing viewpoint. For more details about the flat approximations of surfaces and hypersurfaces see [16,17,29].…”

Normal Developable Surfaces of A Surface Along A Direction Curve

“…Recently, Markina and Raffaelli examined the same topic in R m+1 . Taking a smooth curve γ in an m−dimensional surface M in R m+1 , they gave some results about the existence and uniqueness of a flat surface H having the same field of normal vectors as M along γ [9].…”