GEOMETRIC AND ANALYTIC CONSIDERATIONS OF THE PARALLELISM FOR SUBMANIFOLDS

Abstract

The notions of parallel and self-parallel smooth immersions have been introduced by H. R. Farran and S. A. Robertson [9]. A more de­ tailed study of these notions for simply closed curves was done by F. J. Craveiro De Carvalho, and S. A. Robertson in [6] and by B. Wegner in [18]-in-the•general case and in [20] for curves on surfaces.

Our study is an extension to these investigations and an explicit de­ scription of the connection of parallelism to the so-called total normal twist of closed and non-closed curves.

In the first chapter we expose and investigate some remarks done by other authors for parallel immersions in Euclidean n-space, and its re­ lated notations, especially parallel rank. We give some examples on global parallel rank, covering parallel rank, uniform local parallel rank and local parallel rank satisfying a relation joining these parallel ranks. For plane curves some more detailed results obtained, referring to the connection between parallel rank, curvature and the evolute of the curve. The connection between parallel rank of a smooth immersion of m-dimensional manifold in Euclidean n-space, m < n and the curvature of the normal bundle is explained in detail. The behaviour of parallel rank for cartesian and diagonal product of immersions is studied. At the end of this chapter we discuss the action of conformal transforma­ tions on parallel sections of the normal bundle and parallel ranks of an immersion in the Euclidean n-space. This leads to the insight, that all notions and results easily can be transferred to spaces of constant curvature.