## Journal of the Mathematical Society of Japan

### Weakly reflective submanifolds and austere submanifolds

#### Abstract

An austere submanifold is a minimal submanifold where for each normal vector, the set of eigenvalues of its shape operator is invariant under the multiplication by $-1$. In the present paper, we introduce the notion of weakly reflective submanifold, which is an austere submanifold with a reflection for each normal direction, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of linear isotropy representations of Riemannian symmetric spaces.

#### Article information

Source
J. Math. Soc. Japan Volume 61, Number 2 (2009), 437-481.

Dates
First available in Project Euclid: 13 May 2009

http://projecteuclid.org/euclid.jmsj/1242220718

Digital Object Identifier
doi:10.2969/jmsj/06120437

Mathematical Reviews number (MathSciNet)
MR2532897

Zentralblatt MATH identifier
05573646

Subjects

#### Citation

IKAWA, Osamu; SAKAI, Takashi; TASAKI, Hiroyuki. Weakly reflective submanifolds and austere submanifolds. J. Math. Soc. Japan 61 (2009), no. 2, 437--481. doi:10.2969/jmsj/06120437. http://projecteuclid.org/euclid.jmsj/1242220718.

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