Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ G$-deformations and some generalizations of H. Weyl's tube theorem


Authors: Oldřich Kowalski and Lieven Vanhecke
Journal: Trans. Amer. Math. Soc. 294 (1986), 799-811
MSC: Primary 53C20
DOI: https://doi.org/10.1090/S0002-9947-1986-0825738-0
MathSciNet review: 825738
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove an invariance theorem for the volumes of tubes about submanifolds in arbitrary analytic Riemannian manifolds under $ G$-deformations of the second order. For locally symmetric spaces or two-point homogeneous spaces we give stronger invariance theorems using only $ G$-deformations of the first order. All these results can be viewed as generalizations of the result of H. Weyl about isometric deformations and the volumes of tubes in spaces of constant curvature. They are derived from a new formula for the volume of a tube about a submanifold.


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

  • [1] Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb., Band 93, Springer-Verlag, Berlin, Heidelberg and New York, 1978. MR 496885 (80c:53044)
  • [2] R. B. Brown and A. Gray, Riemannian manifolds with holonomy group Spin(9), Differential Geometry (in honor of K. Yano), Kinokuniya, Tokyo, 1972, pp. 41-59. MR 0328817 (48:7159)
  • [3] B. Y. Chen and L. Vanhecke, Differential geometry of geodesic spheres, J. Reine Angew. Math. 325 (1981), 28-67. MR 618545 (82m:53038)
  • [4] A. Gray and L. Vanhecke, The volumes of tubes in a Riemannian manifold, Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), 1-50. MR 706043 (84i:53053)
  • [5] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. MR 0145455 (26:2986)
  • [6] G. R. Jensen, Deformation of submanifolds of homogeneous spaces, J. Differential Geom. 16 (1981), 213-246. MR 638789 (83h:53063)
  • [7] S. Kobayashi and K. Nomizu, Foundations of differential geometry. I, Interscience, New York, 1963. MR 0152974 (27:2945)
  • [8] O. Kowalski and L. Vanhecke, $ G$-deformations of curves and volumes of tubes in Riemannian manifolds, Geom. Dedicata 15 (1983), 125-135. MR 737953 (85i:53048)
  • [9] F. Tricerri and L. Vanhecke, Geodesic spheres and naturally reductive homogeneous spaces, Proc. Convegno Internazionale di Geometria Differenziale e Analisi Complessa (in honor of E. Martinelli), Roma, 1983, Riv. Mat. Univ. Parma 10 (1984), 123-131. MR 777319 (86b:53049)
  • [10] L. Vanhecke, A note on harmonic spaces, Bull. London Math. Soc. 13 (1981), 545-546. MR 634597 (82m:53025)
  • [11] -, The canonical geodesic involution and harmonic spaces, Ann. Global Anal. Geom. 1 (1983), 131-136. MR 739895 (85k:53022)
  • [12] L. Vanhecke and T. J. Willmore, Interaction of tubes and spheres, Math. Ann. 263 (1983), 31-42. MR 697328 (85c:53085)
  • [13] H. Weyl, On the volume of tubes, Amer. J. Math. 61 (1939), 461-472. MR 1507388

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C20

Retrieve articles in all journals with MSC: 53C20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0825738-0
Keywords: Volumes of tubes about submanifolds, Jacobi vector fields, $ G$-deformations of order $ k$, locally $ k$-equivalent submanifolds, total mean curvatures of tubes
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society