Higher order symmetric spaces and the roots of the identity in a Lie group
HTML articles powered by AMS MathViewer
- by Cecília Ferreira and Armando Machado PDF
- Proc. Amer. Math. Soc. 128 (2000), 2181-2186 Request permission
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 $G$.References
- F. E. Burstall, Harmonic tori in spheres and complex projective spaces, J. Reine Angew. Math. 469 (1995), 149–177. MR 1363828, DOI 10.1515/crll.1995.469.149
- Francis E. Burstall and John H. Rawnsley, Twistor theory for Riemannian symmetric spaces, Lecture Notes in Mathematics, vol. 1424, Springer-Verlag, Berlin, 1990. With applications to harmonic maps of Riemann surfaces. MR 1059054, DOI 10.1007/BFb0095561
- Cecília Ferreira, Embedding flag manifolds of a Hermitian space $E$ into the unitary group $\textrm {U}(E)$, Boll. Un. Mat. Ital. B (7) 7 (1993), no. 3, 575–590 (English, with Italian summary). MR 1244408
- C. Ferreira & A. Machado: Some embeddings of the space of partially complex structures. Portugal. Math. 55 (1998), 485–504.
- Alfred Gray, Riemannian manifolds with geodesic symmetries of order $3$, J. Differential Geometry 7 (1972), 343–369. MR 331281
- Karen Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), no. 1, 1–50. MR 1001271
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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2181-2186
- MSC (1991): Primary 22E15; Secondary 53C30, 53C35
- DOI: https://doi.org/10.1090/S0002-9939-99-05240-5
- MathSciNet review: 1653453