Projections of hypersurfaces in \(\mathbb R^4\) with boundary to planes. (English) Zbl 1290.53007
The authors study the singularities of orthogonal projections of a generic embedded hypersurface \(M\) in \(\mathbb R^4\) with boundary to a two-dimensional plane. The singularities occurring at interior points have been classified in [A. C. Nabarro, Lect. Notes Pure Appl. Math. 232, 283–299 (2003; Zbl 1079.58030)]. In this paper, the authors are concerned with the classification and geometric interpretation of the projections at boundary points. They classify simple map germs from \(\mathbb R^3\) to the plane of codimension less than or equal to \(4\) with the source containing a distinguished plane which is preserved by coordinate changes. They also investigate the geometrical properties of the normal forms of this classification in order to recognize the cases of codimension less than or equal to \(1\).
Reviewer: Maria Aparecida Soares Ruas (São Carlos)
MSC:
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 58K40 | Classification; finite determinacy of map germs |
| 58K05 | Critical points of functions and mappings on manifolds |
Citations:
Zbl 1079.58030References:
| [1] | DOI: 10.1112/blms/13.6.481 · Zbl 0451.58009 · doi:10.1112/blms/13.6.481 |
| [2] | DOI: 10.1007/978-1-4612-5154-5 · doi:10.1007/978-1-4612-5154-5 |
| [3] | Rieger, Compositio Math. 79 pp 99– (1991) |
| [4] | DOI: 10.1112/jlms/s2-36.2.351 · Zbl 0639.58007 · doi:10.1112/jlms/s2-36.2.351 |
| [5] | Nuño-Ballesteros, Proc. Royal Soc. Edinb. 137A pp 1313– (2007) · Zbl 1137.53004 · doi:10.1017/S0308210506000758 |
| [6] | DOI: 10.1112/S1461157000000280 · Zbl 0954.58030 · doi:10.1112/S1461157000000280 |
| [7] | DOI: 10.1017/S0013091502000883 · Zbl 1041.58016 · doi:10.1017/S0013091502000883 |
| [8] | DOI: 10.1017/S0308210500030821 · Zbl 0830.58003 · doi:10.1017/S0308210500030821 |
| [9] | Nabarro, Projections of hypersurfaces in \(\mathbb{R}\)4 to planes, Lecture Notes in Pure and Applied Maths pp 283– (2003) · Zbl 1079.58030 |
| [10] | Damon, Mem. Amer. Math. Soc. 50 pp 1– (1984) |
| [11] | DOI: 10.1007/BF01265348 · Zbl 0834.53006 · doi:10.1007/BF01265348 |
| [12] | Bruce, Quart. J. Math. 49 pp 433– (1998) · doi:10.1093/qmathj/49.4.433 |
| [13] | DOI: 10.1007/BF02684889 · Zbl 0202.55102 · doi:10.1007/BF02684889 |
| [14] | DOI: 10.1088/0951-7715/10/1/017 · Zbl 0929.58019 · doi:10.1088/0951-7715/10/1/017 |
| [15] | DOI: 10.1007/s10711-008-9265-x · Zbl 1158.58019 · doi:10.1007/s10711-008-9265-x |
| [16] | DOI: 10.1112/plms/s3-60.2.392 · Zbl 0667.58002 · doi:10.1112/plms/s3-60.2.392 |
| [17] | DOI: 10.1007/BF01391830 · Zbl 0596.58005 · doi:10.1007/BF01391830 |
| [18] | DOI: 10.1007/978-94-010-0834-1_1 · doi:10.1007/978-94-010-0834-1_1 |
| [19] | DOI: 10.1112/jlms/s2-44.1.155 · Zbl 0767.53004 · doi:10.1112/jlms/s2-44.1.155 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
