Ivancevic_Applied-Diff-Geom

Introduction 11of curves include circles, hyperbolas, and the trefoil knot. Sphere, cylinder,torus, projective plane, 9 Möbius strip, 10 and Klein bottle 11 are examplesof surfaces.Manifolds inherit many of the local properties of Euclidean space. Inparticular, they are locally path–connected, locally compact and locallymetrizable. Being locally compact Hausdorff spaces, they are necessarilyTychonoff spaces. Requiring a manifold to be Hausdorff may seem strange;it is tempting to think that being locally homeomorphic to a Euclideanspace implies being a Hausdorff space. A counterexample is created bydeleting zero from the real line and replacing it with two points, an openneighborhood of either of which includes all nonzero numbers in some openinterval centered at zero. This construction, called the real line with twoorigins is not Hausdorff, because the two origins cannot be separated.All compact surfaces are homeomorphic to exactly one of the 2–sphere,a connected sum of tori, or a connected sum of projective planes.A topological space is said to be homogeneous if its homeomorphismgroup acts transitively on it. Every connected manifold without boundaryis homogeneous, but manifolds with nonempty boundary are not homogeneous.It can be shown that a manifold is metrizable if and only if it is paracompact.Non–paracompact manifolds (such as the long line) are generally9 Begin with a sphere centered on the origin. Every line through the origin piercesthe sphere in two opposite points called antipodes. Although there is no way to do sophysically, it is possible to mathematically merge each antipode pair into a single point.The closed surface so produced is the real projective plane, yet another non-orientablesurface. It has a number of equivalent descriptions and constructions, but this routeexplains its name: all the points on any given line through the origin projects to thesame ‘point’ on this 1plane’.10 Begin with an infinite circular cylinder standing vertically, a manifold withoutboundary. Slice across it high and low to produce two circular boundaries, and thecylindrical strip between them. This is an orientable manifold with boundary, uponwhich ‘surgery’ will be performed. Slice the strip open, so that it could unroll to becomea rectangle, but keep a grasp on the cut ends. Twist one end 180 deg, making the innersurface face out, and glue the ends back together seamlessly. This results in a strip witha permanent half–twist: the Möbius strip. Its boundary is no longer a pair of circles,but (topologically) a single circle; and what was once its ‘inside’ has merged with its‘outside’, so that it now has only a single side.11 Take two Möbius strips; each has a single loop as a boundary. Straighten out thoseloops into circles, and let the strips distort into cross–caps. Gluing the circles togetherwill produce a new, closed manifold without boundary, the Klein bottle. Closing thesurface does nothing to improve the lack of orientability, it merely removes the boundary.Thus, the Klein bottle is a closed surface with no distinction between inside and outside.Note that in 3D space, a Klein bottle’s surface must pass through itself. Building a Kleinbottle which is not self–intersecting requires four or more dimensions of space.