Evolute of a nonregular curve in a space with affine connection

Abstract

One of the basic concepts of the geometry of spaces with affine connection is the concept of parallel transport of a vector along a curve. It is a special case of the general concept of parallel transport along a path in a fibered manifold with connection or, what is the same, the concept of lift of a path into a fibered manifold. Usually these concepts are defined for piecewise differentiable curves, it is shown in [i, 2] how one can naturally extend the concept of lift to some classes of paths containing essentially nonregular curves (for example, all curves of class C ~, ~ > i/2). The problem of extending the concept of parallel transport of a vector along a curve on a surface to the case when the curve belongs to the class C ~ with ~ > 1/2 is also considered in [3, 4]. In [5] the concept of parallel transport along curves of class C ~ with ~ > i/2 in Riemannian spaces is considered. The approach adopted in [3-5] is based on some concrete constructions and in this respect differs from that which is used in [i, 2]. The method of extending parallel transport to the case of nonregular curves adopted in [I, 2] can be characterized as axiomatic and briefly is as follows. One introduces a class K of curves in a differentiable manifold. For curves in this class one defines a concept of convergence. The class K contains curves satisfying the regularity conditions used in differential geometry. It is shown in [i, 2] that the concept of lift can be extended to the class K so that the lift of a curve of class K is a curve of the same class and if curves converge in the class K to a curve, then their lifts converge to the lift of the limit curve in the sense of convergence defined in K. In [i, 2] a concrete class K = K(~) is considered where ~ :[0, ~)-~R is a nondecreasing function. In particular, as m one can take the function t I/2+e where s > 0 and in this case the class K(w) consists of curves satisfying a H~ider condition with exponent 1/2 + e.