Advertisement

Congruence theorems for compact hypersurfaces of a riemannian manifold

  • 20 Accesses

Summary

Let Mm,\(\bar M\) m be two m-dimensional compact oriented hypersurfaces of class C3 immersed in a Riemannian manifold Rm+1 of constant sectional curvature. Suppose that Rm+1 admits a one-parameter continuous group G of conformal transformations satisfying a certain condition (which holds automatically when G is a group of isometric transformations). Suppose further that there is a1 − 1 transformation Tτ ∈ G between Mm and\(\bar M\) m such that\(\bar P = T_{\tau (P)} P\) for each P ∈ Mm and each\(\bar P \in \bar M\) m. If the r-th mean curvature for any r, 1 ⩽ r ⩽ m, of Mm at each point P ∈ Mm is equal to that of\(\bar M\) m at the corresponding point\(\bar P = T_{\tau (P)} P\), together with other conditions, then Mm and\(\bar M\) m are congruent mod G. This is a generalization of a joint theorem ofH. Hopf andY. Katsurada [5] in which G is a group of isometric transformations.

Bibliography

  1. [1]

    A. Aeppli,Einige Ähnlichkeits- und Symmetrie-sätze für differenzierbare Flächen im Raum, Comment. Math. Helv.,33 (1959), pp. 174–195.

  2. [2]

    L. P. Eisenhart,Riemannian geometry, Princeton University Press, Princeton, 1949.

  3. [3]

    H. Hopf -K. Voss,Ein Satz aus der Flächentheorie im Grossen, Arch. Math.,3 (1952), pp. 187–192.

  4. [4]

    H. Hopf -Y. Kutsurada,Some congruence theorems for closed hypersurfaces in Riemann spaces. — II:Method based on a maximum principle, Comment. Math. Helv.,43 (1968), pp. 217–223.

  5. [5]

    H. Hopf -Y. Katsurada,Some congruence theorems for closed hypersurfaces in Riemann spaces. — III:Method based on Voss' proof, Comment. Math. Helv.,46 (1971), pp. 478–486.

  6. [6]

    C. C. Hsiung,Some global theorems on hypersurfaces, Canad. J. Math.,9 (1957), pp. 5–14.

  7. [7]

    C. S. Hsü,Characterization of some elementary transformations, Proc. Amer. Math. Soc.,10 (1959), pp. 324–328.

  8. [8]

    Y. Katsurada,Some congruence theorems for closed hypersufaces in Riemann spaces. — I:Method based on Stokes' theorem, Comment. Math. Helv.,43 (1968), pp. 176–194.

  9. [9]

    R. E. Stong,Some global properties of hypersurfaces, Proc. Amer. Math. Soc.,11 (1960), pp. 126–131.

  10. [10]

    A. W. Tucker,On generalized covariant differentiation, Ann. of Math.,32 (1931), pp. 451–460.

  11. [11]

    K. Voss,Einige differentialgeometrische Kongruenzsätze für geschlossene Flächen und Hyperflächen, Math. Ann.,131 (1956), pp. 180–218.

  12. [12]

    C. E. Weatherburn,An introduction to Riemannian geometry and the tensor calculus, Cambridge University Press, Cambridge, 1950.

Download references

Author information

Additional information

Entrata in Redazione il 13 Giugno 1975.

The first author was partially supported by the National Science Foundation grant GP-33944.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hsiung, C., Lo, T.P. Congruence theorems for compact hypersurfaces of a riemannian manifold. Annali di Matematica 109, 289–304 (1976). https://doi.org/10.1007/BF02416965

Download citation

Keywords

  • Riemannian Manifold
  • Sectional Curvature
  • Conformal Transformation
  • Continuous Group
  • Constant Sectional Curvature