Tohoku Mathematical Journal

Examples of austere orbits of the isotropy representations for semisimple pseudo-Riemannian symmetric spaces

Kurando Baba

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Harvey-Lawson and Anciaux introduced the notion of austere submanifolds in pseudo-Riemannian geometry. We give an equivalent condition for an orbit of the isotropy representations for semisimple pseudo-Riemannian symmetric space to be an austere submanifold in a pseudo-sphere in terms of restricted root system theory with respect to Cartan subspaces. By using the condition we give examples of austere orbits.

Article information

Tohoku Math. J. (2), Volume 71, Number 3 (2019), 397-424.

First available in Project Euclid: 18 September 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53C35: Symmetric spaces [See also 32M15, 57T15]

austere submanifold pseudo-Riemannian symmetric space s-representation restricted root system


Baba, Kurando. Examples of austere orbits of the isotropy representations for semisimple pseudo-Riemannian symmetric spaces. Tohoku Math. J. (2) 71 (2019), no. 3, 397--424. doi:10.2748/tmj/1568772178.

Export citation


  • H. Anciaux, Minimal submanifolds in pseudo-Riemannian geometry, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
  • K. Baba, Local orbit types of the isotropy representations for semisimple pseudo-Riemannian symmetric spaces, Differential Geom. Appl. 38 (2015), 124–150.
  • M. Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. 74 (1957), 85–177.
  • J. Hahn, Isotropy representations of semisimple symmetric spaces and homogeneous hypersurfaces, J. Math. Soc. Japan 40 (1988), 271–288.
  • R. Harvey and H. B. Lawson, Calibrated geometries, Acta Math. 148, (1982), 47–157.
  • S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 2001.
  • O. Ikawa, The geometry of symmetric triad and orbit spaces of Hermann actions, J. Math. Soc. Japan 63 (2011), 79–136.
  • O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds and austere submanifolds, J. Math. Soc. Japan 61 (2009), 437–481.
  • N. Koike, Examples of certain kind of minimal orbits of Hermann actions, Hokkaido Math. J. 43 (2014), 21–42.
  • O. Loos, Symmetric spaces. I: General theory, II: Compact spaces and classification, W. A. Benjamin, Inc., New York-Amsterdam, 1969.
  • T. Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979), 331–357.
  • T. Oshima and T. Matsuki, Orbits on affine symmetric spaces under the action of the isotropy subgroups, J. Math. Soc. Japan 32 (1980), 399–414.
  • T. Oshima and J. Sekiguchi, The restricted root system of a semisimple symmetric pair, Adv. Stud. Math. 4 (1984), 433–497.
  • W. Rossmann, The structure of semisimple symmetric spaces, Canad. J. Math. 31 (1979), 157–180.
  • M. Takeuchi, Modern spherical functions, Translations of Mathematical Monographs, 135, American Mathematical Society, Providence, RI, 1994.
  • G. Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York, 1972.