Manifolds and differential geometry lee pdf


Although manifolds and differential geometry lee pdf manifold locally resembles Euclidean space, meaning that every point has a neighborhood homeomorphic to an open subset of Euclidean space, globally it may not: manifolds in general are not homeomorphic to Euclidean space. Manifolds can be equipped with additional structure. A surface is a two dimensional manifold, meaning that it locally resembles the Euclidean plane near each point. Although no individual map is sufficient to cover the entire surface of the globe, any place in the globe will be in at least one of the charts.

Many places will appear in more than one chart. These regions of the globe will be described in full in separate charts, which in turn will contain parts of North America. Describing the coordinate charts on surfaces explicitly requires knowledge of functions of two variables, because these patching functions must map a region in the plane to another region of the plane. Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle. Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Figure 2: A circle manifold chart based on slope, covering all but one point of the circle. The top, bottom, left, and right charts show that the circle is a manifold, but they do not form the only possible atlas.