Topology

Basic topology by M.A. Armstrong

By M.A. Armstrong

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The geodesic lines included in the system form a one-parameter family of lines on the surface of the obstacle. The straight lines tangent to these geodesics, form a 2-dimensional variety in the 4-dimensional manifold of all straight lines 'of the 3-space. This variety has singular points. These singular points are the unfurled swallowtails (for a typical obstacle problem, that is for a generic obstacle surface and a generic initial condition). The straight lines corresponding to the cusped edge of the unfurled swallowtail, have an asymptotical direction at the point of tangency with the obstacle surface.

Let us consider a generic surface with a generic boundary in Euclidean 3-space. The boundary may be tangent to the curvature lines of the surface at some isolated points of the boundary. I. ARNOLD consists of the 3 parts: 1) the surface curvature centre set (the envelope of the normal lines). 2) the boundary curve curvature centre set (the envelope of its normal planes). 3) the union of the normals of the surface at the points of the boundary curve. The union of these three surfaces is locally diffeomorphic to the caustic of F4 (at a neighbourhood of the curvature centre on the normal to the surface at the point of tangency of the boundary with the curvature line), the surface and the boundary being generic (LG.

But a generic curve may contain some isolatedflaitening points. In a neighbourhood of such a point, the equation of the curve may be written (in suitable affine coordinates) in the form y = x 2 + ... , z = x 4 + ... , where the dots signify the higher order terms. Let us consider now. all the tangent lines of our curve. This developable surface has a semicubical cusped edge - the initial curve. P. Shcherbak [9]). The transition to the curves in higher-dimensional spaces yields higherdimensional folded umbrellas (in the neighbourhood of the "flattening" point the curve in an n-space admits a representation .

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