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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)].
##### MSC:
 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