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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Higher order symmetric spaces
and the roots of the identity in a Lie group


Authors: Cecília Ferreira and Armando Machado
Journal: Proc. Amer. Math. Soc. 128 (2000), 2181-2186
MSC (1991): Primary 22E15; Secondary 53C30, 53C35.
Published electronically: November 29, 1999
MathSciNet review: 1653453
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Abstract: Let $r_k(G)$ denote the set of all $k$-roots of the identity in a Lie group $G$. We show that $r_k(G)$ is always an embedded submanifold of $G$, having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of $k$-symmetric spaces and we show, more generally, that every $k$-symmetric space of a Lie group $G$ is a covering manifold of an embedded submanifold $Orb$ of $G$. We compute also the Hessian of the inclusions of $r_k(G)$ and $Orb$ into $G$, relative to the natural connection on the domain and to the symmetric connection on $G$.


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Additional Information

Cecília Ferreira
Affiliation: CMAF da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
Email: cecilia@lmc.fc.ul.pt

Armando Machado
Affiliation: CMAF da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
Email: armac@lmc.fc.ul.pt

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05240-5
PII: S 0002-9939(99)05240-5
Keywords: Lie group, orbit, $k$-symmetric space, $k$-root of the identity.
Received by editor(s): April 24, 1998
Received by editor(s) in revised form: August 24, 1998
Published electronically: November 29, 1999
Additional Notes: This work was supported by FCT, PRAXIS XXI, FEDER and project PRAXIS/2/ 2.1/MAT/125/94.
Communicated by: Roe Goodman
Article copyright: © Copyright 2000 American Mathematical Society