Smooth Manifolds

Abstract

Because manifolds are locally like euclidean space of some dimension they may inherit local structures from euclidean space. Our first local structure is smoothness: to determine whether a function between euclidean spaces is differentiable we need only investigate what happens in a neighbourhood of each point. By using a chart to transfer the local coordinate structure from euclidean space to a manifold we may use these transferred coordinates to declare a function between manifolds to be differentiable or not. Of course it is necessary to ensure that the answer at a particular point is independent of the chart chosen and that is what we discuss first. After that we describe one of Nyikos’s constructions of differential structures on the long line. We discuss briefly Nyikos’s construction of $2^{\aleph _1}$ many distinct differential structures on the long line, so, in contrast to the situation for metrisable manifolds, non-metrisable manifolds of low dimension support many distinct differential structures. We then describe exotic differential structures on the long plane; again this is in contrast to the metrisable situation where one must wait until dimension four before finding exotic structures, and even then there are only $\mathfrak c$ many of them.