Proceedings of the American Mathematical Society

Sufficient conditions for one domain to contain another in a space of constant curvature

Author(s): Jiazu Zhou
Journal: Proc. Amer. Math. Soc. 126 (1998), 2797-2803.
MSC (1991): Primary 52A22, 53C65; Secondary 51M16
Retrieve article in: PDF
This article is available free of charge

Abstract: As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature $\epsilon$ , that is, Euclidean space ${\mathbb{R}}^{3}$ ,

-sphere $S^{3}$ , hyperbolic space ${\mathbb{H}}^{3}$ ( $\epsilon =0,\,+1,\,-1$ , respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space $\mathbb{R}^{3}$ , or a

-sphere $S^{3}$ or a hyperbolic space $\mathbb{H}^{3}$ .

1.: L. A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, Reading, MA (1976). MR 55:6340
2.: S. S. Chern and Chih-Ta Yen, Formula principale cinematica dello spazio ad n dimensioni, Boll. Un. Mat. Ital. 2 (1940), 434-437. MR 3:89b
3.: Delin Ren, Topics in Integral Geometry, Singapore World Scientific International Publisher (1992). MR 96h:53087
4.: C-S. Chen, On the kinematic formula of square of mean curvature, Indiana Univ. Math. J. 22 (1972-73), 1163-1169. MR 47:2529
5.: G. Zhang, A sufficient condition for one convex body containing another, Chin. Ann. of Math. 9B(4) (1988), 447-451. MR 90k:52008
6.: R. Howard, The kinematic formula in riemannian geometry, Memoir of the Amer. Math. Soc. 509 (1993).
7.: Michael Spivak, A Comprehensive Introduction to Differential Geometry (II), Publish or Perish, Inc. (1979). MR 82g:53003b
8.: H. Hadwiger, Genenseitige Bedeckbarkeit zweier Eibereiche und Isoperimetrie, Viertejsch. Naturforsch. Gesellsch. Zürich 86 (1941), 152-156. MR 4:112c
9.: H. Hadwiger, Überdeckung ebener Bereiche derch Kreise und Quadrate, Comment. Math. Helv. 13 (1941), 195-200. MR 3:90f
10.: Yu. D. Burago & V. A. Zalgaller, Geometric Inequalities, Springer-Verlag Berlin Heidelberg (1988). MR 89b:52020
11.: E. Teufel, On the total absolute curvature of closed curves in spheres, Manuscripta Mathematical 57 (1986), 101-108. MR 88b:53080
12.: Eric Grinberg, Delin Ren & Jiazu Zhou, The isoperimetric inequality and the containment problem in the plane of constant curvature, submitted.
13.: B-Y Chen, Geometry of Submanifolds, Marcel Dekker Inc. New York (1973). MR 50:5697
14.: Jiazu Zhou, Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger's theorem in $R^{2n}$ , Trans. Amer. Math. Soc. 345 no. 1 (1994), 243-262. MR 95a:52009
15.: Jiazu Zhou, The sufficient condition for a convex body to contain another in $R^{4}$ , Proc. Amer. Math. Soc. 121 no. 3 (1994), 907-913. MR 94i:52007
16.: Jiazu Zhou, When can one domain enclose another in $R^{3}$ ?, J. Austral. Math. Soc. (Series A) 59 (1995), 266-272. MR 96f:52008
17.: Jiazu Zhou, A kinematic formula and analogues of Hadwiger's theorem in space, Cont. Math. (Amer. Math. Soc.) 140 (1992), 159-167. MR 93k:52003
18.: F. Brickell & C. C. Hsiung, The total absolute curvature of closed curves in riemannian manifolds, J. Diff. Geom. 9 (1974), 177-193. MR 49:3795
19.: Y. Tsukamoto, On the total absolute curvature of closed curves in manifolds of negative curvature, Math. Ann. 210 (1974), 313- 319. MR 51:1670

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 52A22, 53C65, 51M16

Jiazu Zhou
Affiliation: Department of Mathematics, Sultan Qaboos University, P.O.Box 36, Al-Khod 123, Sultanate of Oman
Address at time of publication: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015-3174
Email: jiz3@lehigh.edu

DOI: 10.1090/S0002-9939-98-04369-X
PII: S 0002-9939(98)04369-X
Keywords: Kinematic formula, transfer principle, Weingarden transformation, Gaussian curvature, convex body, domain, mean curvature, total geodesic curvature.
Received by editor(s): April 25, 1996
Received by editor(s) in revised form: February 18, 1997
Communicated by: Christopher B. Croke
Copyright of article: Copyright 1998, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS