manuscripta mathematica

, Volume 39, Issue 2–3, pp 201–218 | Cite as

Total curvature of manifolds in self-immersed manifolds

  • Toru Ishihara


Chern-Lashof [3] and Kuiper [5] showed the total absolute curvature of a manifold in Euclidean space equals the mean value of the number of critical points of height functions. Teufel [10] proved that a similar result holds for the total absolute curvature of a manifold in a unit sphere. The purpose of this paper is to extend Teufel's result to a relation between the total absolute curvature of some manifolds in self-immersed manifolds and the mean value of the number of zeros of certain vector fields.


Vector Field Fundamental Form Measure Zero Local Coordinate System Height Function 
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  1. [1]
    ALEXANDER, S: Reducibility of Euclidean immersions of low codimension. J. Differential Geometry 3,69–82(1969)MathSciNetzbMATHGoogle Scholar
  2. [2]
    CHERN, S.S.: On the kinematic formula in integral geometry. J. Math. and Mech. 16, 101–118 (1967)MathSciNetzbMATHGoogle Scholar
  3. [3]
    CHERN, S.S, LASHOF, R.K.: On the total curvature of immersed manifolds I. Amer. J. Math. 79,306–318(1957)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    FERUS, D: Totale Absolutkrümmung in Differential-geometrie und- topologie. Lecture Notes in Math. 66, Berlin-Heiderberg-New York: Springer 1968Google Scholar
  5. [5]
    KUIPER, N.H.: Immersions with minimal total absolute curvature. Coll. Geom. Diff. Glob., CBRM 1958, 75–88Google Scholar
  6. [6]
    ISHIHARA,T.: The harmonic Gauss map in a generalized sense.To appear in J. London Math. Soc.Google Scholar
  7. [7]
    MILNOR, J.W.: Topology from the differential view point. The University Press of Virginia, Virginia 1965Google Scholar
  8. [8]
    MOORE, J.D.: Isometric immersions of Riemannian products. J. Differential Geometry 5, 159–168(1971)MathSciNetzbMATHGoogle Scholar
  9. [9]
    MOORE, J.D.: Reducibility of isometric immersions. Proc. Amer. Math. Soc. 34, 229–232(1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    TEUFEL, E.: Eine differentialtopologische Berechnung der totalen Krümmung und totalen Absolutkrümmung in der sphärischen Differentialgeometrie. Manuscripta Math. 31, 119–147 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Toru Ishihara
    • 1
  1. 1.Mathematical Department of Tokushima UniversityTokushimaJapan

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