Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-04T18:59:21.379Z Has data issue: false hasContentIssue false

Any 2-Sphere in E3 with Uniform Interior Tangent Balls is Flat

Published online by Cambridge University Press:  20 November 2018

R. J. Daverman
Affiliation:
The University of Tennessee, Knoxville, Tennessee
L. D. Loveland
Affiliation:
Utah State University, Logan, Utah
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper addresses some flatness properties of an (n – 1)-sphere Σ in Euclidean n-space En resulting from the presence of round balls in En tangent to Σ. The notion of tangency used here is geometric rather than differentiable, for a round n-cell Bp (that is, the set of points whose distance, in the standard metric, from some center point is less than or equal to a fixed positive number) is said to be tangent to the (n – 1)-sphere Σ in En at a point pΣ if pBp and Int BpΣ = ∅. The ball Bp is called an interior tangent ball at p if Int Bp ⊂ Int Σ; otherwise, it is called an exterior tangent ball at p.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Bing, R. H., Approximating surfaces with polyhedral ones, Ann. of Math. 65 (1957), 456483.Google Scholar
2. Bing, R. H., A surface is tame if its complement is 1–ULC, Trans. Amer. Math. Soc. 101 (1961), 294305.Google Scholar
3. Bing, R. H., A wild surface each of whose arcs is tame, Duke Math. J. 28 (1961), 115.Google Scholar
4. Bing, R. H., Spheres in E3 , Amer. Math. Monthly 71 (1964), 353364.Google Scholar
5. Bothe, H. G., Differenzierbare flächen sind sahm, Math. Nachr. 43 (1970), 161180.Google Scholar
6. Brown, M., Locally flat imbeddings of topological manifolds, Ann. of Math. 75 (1962), 331341.Google Scholar
7. Brown, M., Sets of constant distance from a planar set, Michigan Math. J. 19 (1972), 321323.Google Scholar
8. Cannon, J. W., *–taming sets for crumpled cubes. II: Horizontal sections in closed sets, Trans. Amer. Math. Soc. 161 (1971), 441446.Google Scholar
9. Daverman, R. J., Sewings of closed n-cell-complements, Trans. Amer. Math. Soc, to appear.Google Scholar
10. Daverman, R. J. and Loveland, L. D., Wildness and flatness of codimension one spheres having double tangent balls, Rocky Mountain J. Math., to appear.Google Scholar
11. Eaton, W. T., The sum of solid spheres, Michigan Math. J. 19 (1972), 193207.Google Scholar
12. Ferry, S., When *-boundaries are manifolds, Fund. Math. 90 (1976), 199210.Google Scholar
13. Fox, R. H. and Artin, E., Some wild cells and spheres in three-dimensional space, Ann. of Math. 49 (1948), 979990.Google Scholar
14. Gariepy, R. and Pepe, W. D., On the level sets of a distance function in a Minkowski space, Proc. Amer. Math. Soc. 31 (1972), 255259.Google Scholar
15. Harrold, O. G., Jr., Locally peripherally unknotted surfaces in E3 , Ann. of Math. 69 (1959), 276290.Google Scholar
16. Griffith, H. C., Spheres uniformly wedged between balls are tame in E3 , Amer. Math. Monthly 75 (1968), 767.Google Scholar
17. Loveland, L. D., A surface is tame if it has round tangent balls, Trans. Amer. Math. Soc. 152 (1970), 389397.Google Scholar
18. Loveland, L. D., When midsets are manifolds, Proc. Amer. Math. Soc. 61 (1976), 353360.Google Scholar
19. Loveland, L. D., Crumpled cubes that are finite unions of cells, Houston J. Math. 4 (1978), 223228.Google Scholar
20. Loveland, L. D., Unions of cells with applications to visibility, Proc. Amer. Math. Soc, to appear.Google Scholar
21. McMillan, D. R., Jr., Some topological properties of piercing points, Pacific J. Math. 22 (1967), 313322.Google Scholar
22. Pixley, C. P., On crumpled cubes in S3 which are the finite union of tame 3–cells, Houston J. Math. 4 (1978), 105112.Google Scholar
23. Weill, L. R., A new characterization of tame 2–spheres in E3 , Trans. Amer. Math. Soc. 190 (1974), 243252.Google Scholar