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Curves and surfaces in real projective spaces: An approach to generic projections. (English) Zbl 0695.57019
Singularities, Banach Cent. Publ. 20, 335-351 (1988).
[For the entire collection see Zbl 0653.00009.]
This is an expository paper which includes results up to the time of its presentation in 1985. The paper first studies known results about curves in \({\mathbb{R}}^ 2\), in particular formulas relating inflection points, double points, and singularities. Corresponding results in \({\mathbb{R}}{\mathbb{P}}^ 2\) are then reviewed. The paper then reviews results on curves arising in the following manner: let f: \(S\to N\) be a generic mapping where S is a compact surface withouth boundary and N is a connected surface; then the curve (apparent contour) is the image under f of its critical points. Results, illustrated with nice examples, are explored for \(N={\mathbb{R}}\) and \({\mathbb{R}}{\mathbb{P}}^ 2\). The final section studies surfaces, S, embedded in \({\mathbb{R}}{\mathbb{P}}^ n\). A projection is defined from an (n-3) hyperplane P with the target manifold a projective plane, \(\Sigma\). This is used to define a curve, the apparent contour, \(\Delta\), of the projection. Duals \(\Delta^*\) and \(\Sigma^*\) of \(\Delta\) and \(\Sigma\) are defined and results are stated showing a relation among the homology class \([\Delta^*]\) defined by \(\Delta^*\) in \(H_ 1(\Sigma^*,{\mathbb{Z}}_ 2)\), the genus of S, and the homology class [S] induced by the projection of S in \(H_ 2({\mathbb{R}}{\mathbb{P}}^ 2;{\mathbb{Z}}_ 2)\). These results are also illuminated with nice examples.
Reviewer: G.E.Lang jun

MSC:
57R45 Singularities of differentiable mappings in differential topology
57R42 Immersions in differential topology
57R40 Embeddings in differential topology