Total curvature of manifolds in self-immersed manifolds
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Chern-Lashof  and Kuiper  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  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.
KeywordsVector Field Fundamental Form Measure Zero Local Coordinate System Height Function
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- FERUS, D: Totale Absolutkrümmung in Differential-geometrie und- topologie. Lecture Notes in Math. 66, Berlin-Heiderberg-New York: Springer 1968Google Scholar
- KUIPER, N.H.: Immersions with minimal total absolute curvature. Coll. Geom. Diff. Glob., CBRM 1958, 75–88Google Scholar
- ISHIHARA,T.: The harmonic Gauss map in a generalized sense.To appear in J. London Math. Soc.Google Scholar
- MILNOR, J.W.: Topology from the differential view point. The University Press of Virginia, Virginia 1965Google Scholar