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

Author:
Jiazu Zhou

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2797-2803

MSC (1991):
Primary 52A22, 53C65; Secondary 51M16

DOI:
https://doi.org/10.1090/S0002-9939-98-04369-X

MathSciNet review:
1451838

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

- Luis A. Santaló,
*Integral geometry and geometric probability*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. MR**0433364** - S. S. Chern and Chih-Ta Yen,
*Formula principale cinematica dello spazio ad n dimensioni*, Boll. Un. Mat. Ital.**2**(1940), 434–437. - De Lin Ren,
*Topics in integral geometry*, Series in Pure Mathematics, vol. 19, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. Translated from the Chinese and revised by the author; With forewords by Shiing Shen Chern and Chuan-Chih Hsiung. MR**1336595** - Chang-shing Chen,
*On the kinematic formula of square of mean curvature vector*, Indiana Univ. Math. J.**22**(1972/73), 1163–1169. MR**313977**, DOI https://doi.org/10.1512/iumj.1973.22.22096 - Gao Yong Zhang,
*A sufficient condition for one convex body containing another*, Chinese Ann. Math. Ser. B**9**(1988), no. 4, 447–451. A Chinese summary appears in Chinese Ann. Math. Ser. A 9 (1988), no. 5, 635. MR**998651** - R. Howard,
*The kinematic formula in riemannian geometry*, Memoir of the Amer. Math. Soc.**509**(1993). - Michael Spivak,
*A comprehensive introduction to differential geometry. Vol. I*, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR**532830** - H. Hadwiger,
*Genenseitige Bedeckbarkeit zweier Eibereiche und Isoperimetrie*, Viertejsch. Naturforsch. Gesellsch. Zürich**86**(1941), 152–156. - H. Hadwiger,
*Überdeckung ebener Bereiche derch Kreise und Quadrate*, Comment. Math. Helv.**13**(1941), 195–200. - Yu. D. Burago and V. A. Zalgaller,
*Geometric inequalities*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR**936419** - Eberhard Teufel,
*On the total absolute curvature of closed curves in spheres*, Manuscripta Math.**57**(1986), no. 1, 101–108. MR**866407**, DOI https://doi.org/10.1007/BF01172493 - Eric Grinberg, Delin Ren & Jiazu Zhou,
*The isoperimetric inequality and the containment problem in the plane of constant curvature*, submitted. - Bang-yen Chen,
*Geometry of submanifolds*, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 22. MR**0353212** - Jia Zu Zhou,
*Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger’s theorem in ${\bf R}^{2n}$*, Trans. Amer. Math. Soc.**345**(1994), no. 1, 243–262. MR**1250829**, DOI https://doi.org/10.1090/S0002-9947-1994-1250829-7 - Jia Zu Zhou,
*The sufficient condition for a convex body to enclose another in ${\bf R}^4$*, Proc. Amer. Math. Soc.**121**(1994), no. 3, 907–913. MR**1184090**, DOI https://doi.org/10.1090/S0002-9939-1994-1184090-4 - Jia Zu Zhou,
*When can one domain enclose another in ${\bf R}^3$?*, J. Austral. Math. Soc. Ser. A**59**(1995), no. 2, 266–272. MR**1346634** - Jia Zu Zhou,
*A kinematic formula and analogues of Hadwiger’s theorem in space*, Geometric analysis (Philadelphia, PA, 1991) Contemp. Math., vol. 140, Amer. Math. Soc., Providence, RI, 1992, pp. 159–167. MR**1197595**, DOI https://doi.org/10.1090/conm/140/1197595 - F. Brickell and C. C. Hsiung,
*The total absolute curvature of closed curves in Riemannian manifolds*, J. Differential Geometry**9**(1974), 177–193. MR**339032** - Yôtarô Tsukamoto,
*On the total absolute curvature of closed curves in manifolds of negative curvature*, Math. Ann.**210**(1974), 313–319. MR**365418**, DOI https://doi.org/10.1007/BF01434285

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

Retrieve articles in all journals with MSC (1991): 52A22, 53C65, 51M16

Additional Information

**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

MR Author ID:
245435

Email:
jiz3@lehigh.edu

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

Article copyright:
© Copyright 1998
American Mathematical Society