Abstract
We study the distribution of closed geodesics on nilmanifolds Γ \ N arising from a 2-step nilpotent Lie algebra \({\mathfrak {N}}\) constructed from an irreducible representation of a compact semisimple Lie algebra \({\mathfrak {g}o}\) on a real finite dimensional vector space U. We determine sufficient conditions on the semisimple Lie algebra \({\mathfrak {g}o}\) for Γ \ N to have the density of closed geodesics property where Γ is a lattice arising from a Chevalley rational structure on \({\mathfrak{N}}\) .
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Berenshtein, A.D., Zelevinskii, A.V.: When is the multiplicity of a weight equal to 1? Funktsional’nyi Analiz i Ego Prilozheniya, 24, no. 4, pp. 1–13 (1990) English translation in Scientific Council of the USSR Academy of Sciences for the Comprehensive Problem of “Cybernetics”
Bröcker T., tom Dieck T.P.: Representations of Compact Lie Groups. Springer, New York (1985)
Corwin L., Greenleaf P.: Representations of Nilpotent Lie Groups and their Applications. Cambridge University Press, Cambridge (1990)
DeCoste, R.: The multiplicity of weights in nonprimitive pairs of weights. Preprint. arXiv:0708.1757v1
DeMeyer L.: Closed geodesics in compact nilmanifolds arising from group representations. Manuscripta Math. 105, 283–310 (2001)
Eberlein P.: Geometry of 2-step nilpotent groups with a left invariant metric, I. Ann. Scient. Ecole Normale Sup. 27, 611–660 (1994)
Eberlein, P.: Rational approximation in compact Lie groups and their Lie algebras. Preprint (2000) http://www.math.unc.edu/Faculty/pbe/QD_I,_12_26.pdf
Eberlein, P.: 2-Step nilpotent Lie algebras arising from semisimple modules. Preprint (2005) arXiv:0806.2844[math.DG]
Eberlein, P.: Geometry of 2-step Nilpotent Lie Groups, Modern Dynamical Systems, Cambridge University Press, Cambridge, pp. 67–101 (2004)
Gornet R., Mast M.: The length spectrum of Riemannian 2-step nilmanifolds. Ann. Scient. Ecole Normale Sup. 33, 181–209 (2000)
Helgason S.: Differential Geometry and Symmetric Spaces. Academic Press, New York (1962)
Helgason S.: Differential Geometry, Lie Groups and Symmetric Spaces. American Mathematical Society, Providence (2001)
Humphreys J.E.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1972)
Kaplan A.: Riemannian nilmanifolds attached to Clifford modules. Geom. Dedicata 11, 127–136 (1981)
Lang S.: Algebra, 3rd edn. Addison-Wesley, Reading (1993)
Lauret J.: Homogeneous nilmanifolds attached to representations of compact Lie groups. Manuscripta Math. 99, 287–309 (1999)
Lee K., Park K.: Smoothly closed geodesics in 2-step nilmanifolds. Indiana Univ. Math. J. 45, 1–14 (1996)
Mal’cev, A.I.: On a class of homogeneous spaces. Am. Math. Soc. Transl. 39 (1951)
Mast M.: Closed geodesics in 2-step nilmanifolds. Indiana Univ. Math. J. 43, 885–911 (1994)
Mirsky L.: An Introduction to Linear Algebra. Clarendon Press, Oxford (1955)
Raghunathan M.S.: Arithmetic lattices in semisimple groups. Proc. Indian Acad. Sci. (Math. Sci.) 91, 133–138 (1982)
Raghunathan M.S.: Discrete Subgroups of Lie Groups. Springer, New York (1972)
Sansuc J.J.: Groupe de Brauer et arithmetique des groupes algebriques lineaires sur un corps de nombres. J. Reine Angew. Math. 327, 12–80 (1981)
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DeCoste, R.C. Closed geodesics on compact nilmanifolds with Chevalley rational structure. manuscripta math. 127, 309–343 (2008). https://doi.org/10.1007/s00229-008-0206-7
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DOI: https://doi.org/10.1007/s00229-008-0206-7