Journal of the Mathematical Society of Japan

Weakly reflective submanifolds and austere submanifolds

Osamu IKAWA, Takashi SAKAI, and Hiroyuki TASAKI

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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

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1242220718

Digital Object Identifier
doi:10.2969/jmsj/06120437

Zentralblatt MATH identifier
05573646

Mathematical Reviews number (MathSciNet)
MR2532897

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

Keywords
reflective submanifold austere submanifold symmetric space $s$-representation $R$-space

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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