## Tohoku Mathematical Journal

### A Cartan type identity for isoparametric hypersurfaces in symmetric spaces

Naoyuki Koike

#### Abstract

In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted (=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore, by using the Cartan type identity, we show that certain kind of curvature-adapted isoparametric hypersurfaces in a symmetric space of non-compact type are principal orbits of Hermann actions.

#### Article information

Source
Tohoku Math. J. (2), Volume 66, Number 3 (2014), 435-454.

Dates
First available in Project Euclid: 8 October 2014

https://projecteuclid.org/euclid.tmj/1412783206

Digital Object Identifier
doi:10.2748/tmj/1412783206

Mathematical Reviews number (MathSciNet)
MR3266740

Zentralblatt MATH identifier
1306.53048

Subjects

#### Citation

Koike, Naoyuki. A Cartan type identity for isoparametric hypersurfaces in symmetric spaces. Tohoku Math. J. (2) 66 (2014), no. 3, 435--454. doi:10.2748/tmj/1412783206. https://projecteuclid.org/euclid.tmj/1412783206

#### References

• J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132–141.
• J. Berndt, Real hypersurfaces in quaterionic space forms, J. Reine Angew. Math. 419 (1991), 9–26.
• J. Berndt and L. Vanhecke, Curvature adapted submanifolds, Nihonkai Math. J. 3 (1992), 177–185.
• J. Berndt and M. Brück, Cohomogeneity one actions on hyperbolic spaces, J. Reine Angew. Math. 541 (2001), 209–235.
• J. Berndt, S. Console and C. Olmos, Submanifolds and holonomy, Chapman & Hall/CRC Res. Notes Math.$\setminus$434, Chapman & Hall/CRC, Boca Raton, FL, 2003.
• J. Berndt and H. Tamaru, Homogeneous codimension one foliations on noncompact symmetric space, J. Differential Geom. 63 (2003), 1–40.
• J. Berndt and H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit, Tohoku Math. J. 56 (2004), 163–177.
• J. E. D'Atri, Certain isoparametric families of hypersurfaces in symmetric spaces, J. Differential Geom. 14 (1979), 21–40.
• H. Ewert, Equifocal submanifolds in Riemannian symmetric spaces, Doctoral thesis.
• L. Geatti, Complex extensions of semisimple symmetric spaces, Manuscripta Math. 120 (2006), 1–25.
• O. Goertsches and G. Thorbergsson, On the geometry of the orbits of Hermann actions, Geom. Dedicata 129 (2007), 101–118.
• E. Heintze, X. Liu and C. Olmos, Isoparametric submanifolds and a Chevalley type restriction theorem, Integrable systems, geometry, and topology, 151–190, AMS/IP Stud. Adv. Math. 36, Amer. Math. Soc., Providence, RI, 2006.
• E. Heintze, R. S. Palais, C. L. Terng and G. Thorbergsson, Hyperpolar actions on symmetric spaces, Geometry, topology, & physics, 214–245, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995.
• S. Helgason, Differential geometry, Lie groups and symmetric spaces, Pure Appl. Math. 80, Academic Press, Inc., New York-London, 1978.
• W. Y. Hsiang and H. B. Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geom. 5 (1971), 1–38.
• T. Kimura and M. S. Tanaka, Stability of certain minimal submanifolds in compact symmetric spaces of rank two, Differential Geom. Appl. 27 (2009), 23–33.
• N. Koike, On proper Fredholm submanifolds in a Hilbert space arising from submanifolds in a symmetric space, Japan. J. Math. (N.S.) 28 (2002), 61–80.
• N. Koike, Submanifold geometries in a symmetric space of non-compact type and a pseudo-Hilbert space, Kyushu J. Math. 58 (2004), 167–202.
• N. Koike, Complex equifocal submanifolds and infinite dimensional anti-Kaehlerian isoparametric submanifolds, Tokyo J. Math. 28 (2005), 201–247.
• N. Koike, Actions of Hermann type and proper complex equifocal submanifolds, Osaka J. Math. 42 (2005), 599–611.
• N. Koike, A splitting theorem for proper complex equifocal submanifolds, Tohoku Math. J. 58 (2006), 393–417.
• N. Koike, The homogeneous slice theorem for the complete complexification of a proper complex equifocal submanifold, Tokyo J. Math. 33 (2010), 1–30.
• N. Koike, On curvature-adapted and proper complex equifocal submanifolds, Kyungpook Math. J. 50 (2010), 509–536.
• N. Koike, Hermann type actions on a pseudo-Riemannian symmetric space, Tsukuba J. Math. 34 (2010), 137–172.
• N. Koike, Examples of a complex hyperpolar action without singular orbit, Cubo A Math. 12 (2010), 127–143.
• N. Koike, The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry, arXiv:math.DG/0807.1601v3.
• A. Kollross, A Classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc. 354 (2001), 571–612.
• R. Miyaoka, Transnormal functions on a Riemannian manifold, Differential Geom. Appl. 31 (2013), 130–139.
• T. Murphy, Curvature-adapted submanifolds of symmetric spaces, Indiana Univ. Math. J. 61 (2012), 831–847.
• B. O'Neill, Semi-Riemannian Geometry, with Applications to Relativity, Academic Press, New York, 1983.
• R. S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299–340.
• R. S. Palais and C. L. Terng, Critical point theory and submanifold geometry, Lecture Notes in Math. 1353, Springer-Verlag, Berlin, 1988.
• R. Szöke, Adapted complex structures and geometric quantization, Nagoya Math. J. 154 (1999), 171–183.
• R. Szöke, Involutive structures on the tangent bundle of symmetric spaces, Math. Ann. 319 (2001), 319–348.
• R. Szöke, Canonical complex structures associated to connections and complexifications of Lie groups, Math. Ann. 329 (2004), 553–591.
• Z. Tang, Multiplicities of equifocal hypersurfaces in symmetric spaces, Asian J. Math. 2 (1998), 181–214.
• C. L. Terng and G. Thorbergsson, Submanifold geometry in symmetric spaces, J. Differential Geom. 42 (1995), 665–718.
• Q. M. Wang, Isoparametric functions on Riemannian manifolds I, Math. Ann. 277 (1987), 639–646.
• B. Wu, Isoparametric submanifolds of hyperbolic spaces, Trans. Amer. Math. Soc. 331 (1992), 609–626.