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Normal curvatures and Euler classes for polyhedral surfaces in 4-space. (English) Zbl 0558.57010
Just as the total (tangential) curvature of a surface is the (tangential) Euler characteristic, the total normal curvature of a smooth immersion \(f: M^ 2\to E^ 4\) equals the normal Euler number of f. The first fact can be proved by expressing the total curvature as the average of the number of critical points of the orthogonal projections to \(E^ 1\) and then applying the index theorem. Similarly the second fact can be proved by considering the average number of Whitney pinch points of the orthogonal projections to \(E^ 3\). In a previous paper [J. Differ. Geom. 1, 245-256 (1967; Zbl 0164.229)] the author introduced a concept of counting critical points in the case of a polyhedral surface leading to an analogous version of the Gauß-Bonnet theorem. In the present paper he describes a similar concept for a polyhedral analogue of counting pinch points. This leads to a combinatorial formula for the normal Euler number of polyhedral surfaces in \(E^ 4\). The details are announced to appear in a forthcoming paper (The normal Euler class of a polyhedral surface in 4-space). For the particular case of lattice polyhedra see the paper by B. V. Yusin [Proc. Am. Math. Soc. 92, 578-592 (1984)] reviewed above.
Reviewer: W.Kühnel

MSC:
57R20 Characteristic classes and numbers in differential topology
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
52Bxx Polytopes and polyhedra
57Q35 Embeddings and immersions in PL-topology
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