## Asian Journal of Mathematics

- Asian J. Math.
- Volume 12, Number 4 (2008), 421-442.

### On Non-existenceness of Equifocal Submanifolds with Non-flat Section

#### Abstract

We first prove a certain kind of splitting theorem for an equifocal submanifold with non-flat section in a simply connected symmetric space of compact type, where an equifocal submanifold means a submanifold with parallel focal structure. By using the splitting theorem, we prove that there exists no equifocal submanifold with non-flat section in an irreducible simply connected symmetric space of compact type whose codimension is greater than the maximum of the multiplicities of roots of the symmetric space or the maximum added one. In particular, it follows that there exists no equifocal submanifold with non-flat section in some irreducible simply connected symmetric spaces of compact type and that there exists no equifocal submanifold with non-flat section in simply connected compact simple Lie group whose codimension is greater than two.

#### Article information

**Source**

Asian J. Math., Volume 12, Number 4 (2008), 421-442.

**Dates**

First available in Project Euclid: 20 February 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ajm/1235140166

**Mathematical Reviews number (MathSciNet)**

MR2481684

**Zentralblatt MATH identifier**

05554935

**Subjects**

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

**Keywords**

Equifocal submanifold polar action

#### Citation

Koike, Naoyuki. On Non-existenceness of Equifocal Submanifolds with Non-flat Section. Asian J. Math. 12 (2008), no. 4, 421--442. https://projecteuclid.org/euclid.ajm/1235140166