Tokyo Journal of Mathematics

Theorems of Gauss-Bonnet and Chern-Lashof Types in a Simply Connected Symmetric Space of Non-Positive Curvature

Naoyuki KOIKE

Full-text: Open access


In this paper, we shall generalize the Gauss-Bonnet and Chern-Lashof theorems to compact submanifolds in a simply connected symmetric space of non-positive curvature. Those proofs are performed by applying the Morse theory to squared distance functions because height functions are not defined.

Article information

Tokyo J. Math., Volume 26, Number 2 (2003), 527-539.

First available in Project Euclid: 5 June 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53C40: Global submanifolds [See also 53B25]


KOIKE, Naoyuki. Theorems of Gauss-Bonnet and Chern-Lashof Types in a Simply Connected Symmetric Space of Non-Positive Curvature. Tokyo J. Math. 26 (2003), no. 2, 527--539. doi:10.3836/tjm/1244208606.

Export citation


  • S.S. Chern and R.K. Lashof, On the total curvature of immersed manifolds I, Amer. J. Math., 79 (1957), 306–318.
  • S.S. Chern and R. K. Lashof, On the total curvature of immersed manifolds II, Michigan Math. J., 5 (1958), 5–12.
  • D. Ferus, Totale Absolutkr$\ddot u$mmung in Differentialgeometrie und -topologie, Lecture Notes 66, Springer (1968).
  • S. Helgason, Differential geometry$,$ Lie groups and symmetric spaces, Academic Press (1978).
  • N. Koike, The Lipschitz-Killing curvature for an equiaffine immersion and theorems of Gauss-Bonnet type and Chern-Lashof type, Results Math., 39 (2001), 230–244.
  • N. Koike, The total absolute curvature of an equiaffine immersion, Results Math., 42 (2002), 81–106.
  • N. Koike, Tubes of nonconstant radius in a symmetric space, Kyushu J. of Math., 56 (2002), 267–291.
  • N.H. Kuiper, Minimal total absolute curvature for immersions, Invent. Math., 10 (1970), 209–238.
  • N.H. Kuiper, Tight embeddings and maps. Submanifolds of geometrical class three in $E^n$, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979) Springer (1980), 97–145.
  • J. Milnor, Morse theory, Ann. Math. Stud. 51 (1963), Princeton University Press.
  • K. Nomizu and L. Rodriguez, Umbilical submanifolds and Morse functions, Nagoya Math. J., 48 (1972), 197–201.
  • C.L. Terng and G. Thorbergsson, Submanifold geometry in symmetric spaces, J. Differential Geom., 42 (1995), 665–718.
  • E. Teufel, Differential topology and the computation of total absolute curvature, Math. Ann., 258 (1982), 471–480.
  • E. Teufel, On the total absolute curvature of immersions into hyperbolic spaces, Topic in Differential Geometry Vol II, North-Holland (1988), 1201–1210.
  • T.J. Willmore and B.A. Saleemi, The total absolute curvature of immersed manifolds, J. London Math. Soc., 41 (1966), 153–160.
  • T.J. Willmore, Total curvature in Riemannian geometry, Ellis-Horwood (1982).