Maps generating normals on a manifold

The first- and second-order canonical forms are considered on a smooth m-manifold. The approach connected with coordinate expressions of basic and fiber forms, and first- and second-order tangent vectors on a manifold is implemented. It is shown that the basic first- and secondorder tangent vectors are first- and second-order Pfaffian (generalized) differentiation operators of functions set on a manifold. Normals on a manifold are considered. Differentials of the basic first-order (second-order) tangent vectors are linear maps from the first-order tangent space to the second-order (third-order) tangent space. Differentials of the basic first-order tangent vectors (basic vectors of the first-order normal) set a map which takes a first-order tangent vector into the second-order normal for the first-order tangent space of (into the third-order normal for second-order tangent space). The map given by differentials of the basic first-order tangent vectors takes all tangent vectors to the vectors of the second-order normal. Such splitting the osculating space in the direct sum of the first-order tangent space and second-order normal defines the simplest (canonical) affine connection which horizontal subspace is the indicated normal. This connection is flat and symmetric. The second-order normal is the horizontal subspace for this connection. Derivatives of some basic vectors in the direction of other basic vectors are equal to values of differentials of the first vectors on the second vectors, and the order of differentials is equal to the order of vectors along which differentiation is made. The maps set by differentials of basic vectors of r-order normal for (r – 1)-order tangent space take the basic first-order tangent vectors to the basic vectors of (r + 1)-order normal for r-order tangent space. Horizontal vectors for the simplest second-order connection are constructed. Second-order curvature and torsion tensors vanish in this connection. For horizontal vectors of the simplest second-order connection there is the decomposition by means of the basic vectors of the secondand third-order normals.

On the generalized helices of Hayden and Sypták in an N-space

In this paper we are concerned with curves (C) of the following types:where k1, k2, …, kh−1, kh = 0 (h ≤ n) are the curvatures of (C) relative to the space Vn in which (C) lie. Hayden proved that a curve in a Vn is an (A)2m, h = 2m + 1, if and only if it admits an auto-parallel vector along it which lies in the osculating space of the curve and makes constant angles with the tangent and the principal normals. Independently, Sypták∥ stated without proof that a curve in an Rn is a (B)n if and only if it admits a certain number of fixed R2's having the same angle properties; he also gave to such a curve a set of canonical equations from which many interesting properties follow as immediate consequences.

Generalized Helices in an Ordinary Vn

A linear vector m-space Rm defined along a curve (C) in a Vn and lying in the complete osculating space of (C) will be called a characteristic Rm of (C) if it is auto-parallel along the curve and makes constant angles with its tangent and the principal normals. Curves admitting a characteristic R1 have been studied by Hayden under the name of generalized helices and generalized by me§. In this paper we give a complete determination of the curves with a characteristic R2. Curves whose curvatures are proportional to a set of constants, which have been considered by Syptak for the particular case when Vn is an Rn, form one of the classes of curves of this type. As a consequence, the existence of the curves admitting a characteristic Rm (m > 2) is partly established, but the problem has not been completely solved. At the end we prove two theorems in connexion with two particular types of characteristic Rm's.