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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.2 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000200009 

CUBO A Mathematical Journal Vol.12, N° 02, (127-143). June 2010

 

Examples of a complex hyperpolar action without singular orbit

 

Naoyuki Koike

Department of Mathematics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka Shinjuku-ku, Tokyo 162-8601, Japan email: koike@ma.kagu.tus.ac.jp


ABSTRACT

The notion of a complex hyperpolar action on a symmetric space of non-compact type has recently been introduced as a counterpart to the hyperpolar action on a symmetric space of compact type. As examples of a complex hyperpolar action, we have Hermann type actions, which admit a totally geodesic singular orbit (or a fixed point) except for one example. All principal orbits of Hermann type actions are curvature-adapted and proper complex equifocal. In this paper, we give some examples of a complex hyperpolar action without singular orbit as solvable group free actions and find complex hyperpolar actions all of whose orbits are non-curvature-adapted or non-proper complex equifocal among the examples. Also, we show that some of the examples possess the only minimal orbit.

Key words and phrases: symmetric space, complex hyperpolar action, complex equifocal submanifold.


RESUMEN

La noción de una acción hiperpolar compleja sobre un espacio simétrico de tipo no compacto fue recientemente introducida como el análogo de la acción hiperpolar sobre un espacio simétrico de tipo compacto. Como ejemplos de una acción hiperpolar complejas, nosotros tenemos acciones de tipo Hermann, las cuales admiten una orbita (o un punto fijo) singular totalmente geodesica excepto para un ejemplo. Todas las orbitas principales de acciones de tipo Hermann son curvatura-adaptadas y unifocales complejas propias. este artículo, nosotros damos algunos ejemplos de una acción hiperpolar compleja sin orbitas singulares como grupo soluble de acciones libres y encontramos acciones complejas hiperpolares cuyas orbitas son no curvatura-adaptadas o no propias unifocales complejas. También, mostramos que algunos de los ejemplos poseen solamente orbitas minimales.

AMS (MOS) Subj. Class.: 53C35; 53C40


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Received: September 2008.

Revised: April 2009.