Low-Dimensional Manifolds

Abstract

Modern differential geometry is an independent scientific discipline, exceedingly branched and connected with numerous applications. Today’s edifice of differential geometry exhibits two basic layers. The one which was historically the first to appear may be conditionally called local differential geometry which usually develops in a region of a Euclidean space. This is the foundation and the first storeys of the whole edifice. Then there appeared next storeys which took their shape later than the first ones. They are referred to as “global differential geometry”. Here local concepts are closely interwoven with global, topological ones. The backbone of the whole building is the theory of smooth manifolds. On lower storeys, it is at first local, i.e. events occur in a small enough domain of a manifold. Ascending, the theory develops global aspects and finally, on the top, the modern geometry operates with essentially nonlocal effects.