## Tokyo Journal of Mathematics

### The Gauss-Bonnet and Chern-Lashof Theorems in a Simply Connected Symmetric Space of Compact Type

Naoyuki KOIKE

#### Abstract

In this paper, we prove the theorems of the Gauss-Bonnet and Chern-Lashof types for low dimensional compact submanifolds in a simply connected symmetric space of compact type. In particular, in the case where the ambient space is a sphere, we need not to give the restriction for the dimension of the submanifold. Those proofs are performed by applying the Morse theory to squared distance functions.

#### Article information

Source
Tokyo J. Math., Volume 28, Number 2 (2005), 483-497.

Dates
First available in Project Euclid: 5 June 2009

https://projecteuclid.org/euclid.tjm/1244208203

Digital Object Identifier
doi:10.3836/tjm/1244208203

Mathematical Reviews number (MathSciNet)
MR2191062

Zentralblatt MATH identifier
1100.53057

#### Citation

KOIKE, Naoyuki. The Gauss-Bonnet and Chern-Lashof Theorems in a Simply Connected Symmetric Space of Compact Type. Tokyo J. Math. 28 (2005), no. 2, 483--497. doi:10.3836/tjm/1244208203. https://projecteuclid.org/euclid.tjm/1244208203

#### References

• 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 undtopologie, Lecture Notes 66, Springer-Verlag (1968).
• Ishihara, T., The Euler characteristics and Weyl's curvature invariants of submanifolds in spheres, J. Math. Soc. Japan 39 (1987), 247–256.
• N. Koike, The Gauss-Bonnet and Chern-Lashof theorems in simply connected symmetric space of non-positive curvature, Tokyo J. Math. 26 (2003), 527–539.
• E. Teufel, On the total absolute curvature of closed curves in spheres, manuscripta math. 57 (1986), 101–108.