Loading...
Please wait, while we are loading the content...
Similar Documents
Codimension two isometric immersions between Euclidean spaces.
| Content Provider | Semantic Scholar |
|---|---|
| Author | Whitt, Lee B. |
| Copyright Year | 1985 |
| Abstract | Hartman and Nirenberg showed that any C°° isometric immersion/: E" -> E" +1 between flat Euclidean spaces is a cylinder erected over a plane curve. We show that in the codimension two case, /: E" -» E" +2 factors as a composition of isometric immersions/ ~ f\° fe E" -» E" +1 -* E w+2 , when n > 1 and /has nowhere zero normal curvature. Counterexamples are given if this assumption is relaxed. How can paper be folded? More precisely, how can flat Euclidean 2-space E2 be isometrically immersed into flat Euclidean «-space E" (for simplicity, assume C°° differentiability everywhere). For n = 3, A. V. Pogorelov [4] announced without proof that the image is a cylinder erected over a plane curve; proofs may be found in Massey [3] and Stoker [5]. In this paper, we consider n = 4 and show that any isometric immersion g: E2 -> E4 with nowhere zero normal curvature factors as a composition of isometric immersions g = g1 g2: E2 -> E2 -> E4. The result of Pogorelov has been generalized by Hartman and Nirenberg [2]. They showed that the image of any codimension-one isometric immersion between flat Euclidean spaces is a cylinder erected over a plane curve. Using a result of Hartman [1] we easily show that any codimensiontwo, isometric immersion/: E" -> E"+2, n > 1, with nowhere zero normal curvature factors as a composition / = =/ 1 o / 2: E /ί ->E" +1 -»E" +2 . The images of fx and/2 are cylinders. The assumption of nowhere zero normal curvature is essential; counterexamples are given in §3 when the assumption is relaxed. From another point of view, the cylinders of Pogorelov and Hartman and Nirenberg can be deformed ("unrolled") through a one-parameter family of isometric immersions to a hyperplane. This family is obtained by deforming the generating plane curve to a straight line. From our results, it follows easily that any isometric immersion/: Έn -> En+2 with nowhere zero normal curvature can be deformed through isometric immersions to a standard inclusion i: E π «-» En+2 (it would be interesting to know if the normal curvature assumption can be removed). In addition, we proved [7] that if the normal curvature is identically zero, then any |
| Starting Page | 481 |
| Ending Page | 487 |
| Page Count | 7 |
| File Format | PDF HTM / HTML |
| DOI | 10.2140/pjm.1985.119.481 |
| Alternate Webpage(s) | https://msp.org/pjm/1985/119-2/pjm-v119-n2-p14-s.pdf |
| Alternate Webpage(s) | https://doi.org/10.2140/pjm.1985.119.481 |
| Volume Number | 119 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
