Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



$C^1$-homogeneous compacta in $\mathbb R^n$
are $C^1$-submanifolds of $\mathbb R^n$

Authors: Dusan Repovs, Arkadij B. Skopenkov and Evgenij V. Scepin
Journal: Proc. Amer. Math. Soc. 124 (1996), 1219-1226
MSC (1991): Primary 53A04, 54F65, 26A24; Secondary 26A03, 54F50, 26A16, 28A15
MathSciNet review: 1301046
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give the characterization of $C^1$-homogeneous compacta in $\mathbb R^n$: Let $K$ be a locally compact (possibly nonclosed) subset of $\mathbb R^n$. Then $K$ is $C^1$-homogeneous if and only if $K$ is a $C^1$-submanifold of $\mathbb R^n$.

References [Enhancements On Off] (What's this?)

  • 1. V. I. Arnol$'$d, Ordinary differential equations, Nauka, Moscow, 1971. (Russian) MR 50:13677
  • 2. S. Bochner and D. Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639--653. MR 8:253c
  • 3. G. E. Bredon, Introduction to compact transformation groups, Pure Appl. Math., vol. 46, Academic Press, New York, 1972. MR 54:1265
  • 4. R. J. Daverman and L. D. Loveland, Wildness and flatness of codimension one spheres having double tangent balls, Rocky Mountain J. Math. 11 (1981), 113--121. MR 82m:57010
  • 5. D. Dimovski and D. Repov\v{s}, On homogeneity of compacta in manifolds, Atti. Sem. Mat. Fis. Univ. Modena 43 (1995), 25--31. CMP 95:14
  • 6. D. Dimovski, D. Repov\v{s}, and E. V. \v{S}\v{c}epin, $C^\infty$-homogeneous curves on orientable closed surfaces, Geometry and Topology (G. M. Rassias and G. M. Stratopoulos, eds.), World Singapore (1989), 100--104. MR 91e:57023
  • 7. A. N. Drani\v{s}nikov, On free actions of zero-dimensional compact groups, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 1, 212--228; English transl., Math. USSR-Izv. 32 (1989), 217--232. MR 90e:57065
  • 8. J. Dugundji, Topology, Allyn and Bacon, Boston, (1966). MR 33:1824
  • 9. H. Federer, Geometric measure theory, Grundlehren Math. Wiss., vol. 153, Springer-Verlag, Berlin, 1969. MR 41:1976
  • 10. P. J. Giblin and D. B. O'Shea, The bitangent sphere problem, Amer. Math. Monthly 97 (1990), 5--23. MR 91k:53008
  • 11. H. C. Griffith, Spheres uniformly wedged between balls are tame in $E^3$, Amer. Math. Monthly 75 (1968), 767. MR 38:2753
  • 12. K. Kuratowski, Topology, Vol. 2, Academic Press, New York, (1968). MR 41:4467
  • 13. L. D. Loveland, A surface in $E^3$ is tame if it has round tangent balls, Trans. Amer. Math. Soc. 152 (1970), 389--397. MR 42:5270
  • 14. ------, Double tangent ball embeddings of curves in $E^3$, Pacific J. Math. 104 (1983), 391--399. MR 84e:57015
  • 15. ------, Tangent ball embeddings of sets in $E^3$, Rocky Mountain J. Math. 17 (1987), 141--150. MR 88g:57019
  • 16. ------, Spheres with continuous tangent planes, Rocky Mountain J. Math. 17 (1987), 829--844. MR 89b:57007
  • 17. L. D. Loveland and D. G. Wright, Codimension one spheres in $\mathbb R^n$ with double tangent balls, Topology Appl. 13 (1982), 311--320. MR 83h:57023
  • 18. D. Montgomery and L. Zippin, Topological transformation groups, Interscience Tracts in Pure and Appl. Math., vol. 1, Interscience, New York, 1955. MR 17:383b
  • 19. I. I. Natanson, Theory of functions of a real variable, Gosud. Izdat. Tehn. Liter., Moscow (1957), (Russian). MR 50:7454
  • 20. D. Repov\v{s}, A. B. Skopenkov, and E. V. \v{S}\v{c}epin, A characterization of $C^1$-homogeneous subsets of the plane, Boll. Un. Mat. Ital. Ser. A 7 (1993), 437--444. MR 95e:54045
  • 21. ------, Group actions on manifolds and smooth ambient homogeneity, Proc. Colloq. Geometry (Moscow 1993), Mir, Moscow (to appear).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53A04, 54F65, 26A24, 26A03, 54F50, 26A16, 28A15

Retrieve articles in all journals with MSC (1991): 53A04, 54F65, 26A24, 26A03, 54F50, 26A16, 28A15

Additional Information

Dusan Repovs

Keywords: $C^1$-homogeneous compacta, $C^1$-submanifold of $\mathbb R^n$, Hilbert-Smith conjecture, LIP-homeomorphism, Lipschitz chart, almost everywhere differentiability
Received by editor(s): January 15, 1992
Received by editor(s) in revised form: September 15, 1994
Additional Notes: The first author was supported in part by the Ministry of Science and Technology of the Republic of Slovenia grant No. P1-0214-101-92.
Communicated by: James E. West
Article copyright: © Copyright 1996 American Mathematical Society