zbMATH — the first resource for mathematics

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