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Smooth tight immersions. (English) Zbl 0924.53041
This is a short survey about smooth tight immersions of compact manifolds into Euclidean space. In the Morse-theoretic interpretation, an immersion is called tight if any nondegenerate height function is a perfect function. Section 2 gives the classical bounds on the essential codimension which are due to N. H. Kuiper. Section 3 reports on the case of 2-dimensional surfaces. Here it is worth mentioning that U. Pinkall’s work on tightness for regular homotopy classes of surfaces in \(E^3\) [Topology 25, 475-481 (1986; Zbl 0605.53027)] was later corrected and extended by D. P. Cervone [Topology 35, 863-873 (1996; Zbl 0858.53051)]. In Section 4, the case of highly connected manifolds is discussed. In particular, this includes the recent work of R. Niebergall and the author on the classification of such tight immersions \(M^{2k}\to E^{3k+2}\). A number of open problems are mentioned.
In addition to this survey, the reader may consult the recent volume [‘Tight and taut submanifolds’ (Berkeley, CA, 1994)(Cambridge University Press) (1997; Zbl 0883.00009)].
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
57R70 Critical points and critical submanifolds in differential topology
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry