Geodesic orbit metrics in compact homogeneous manifolds with equivalent isotropy submodules. (English) Zbl 1404.53054

The author studies the classification of GO-metrics on compact homogeneous manifolds \(M\cong G/H\).
Using a result based on a proposition by D. Alekseevsky and A. Arvanitoyeorgos [Trans. Am. Math. Soc. 359, No. 8, 3769–3789 (2007; Zbl 1148.53038)], the problem of finding G-GO metrics on \(M\) is reduced to the algebraic problem of determining metric endomorphisms \(A:\mathfrak{m}\rightarrow \mathfrak{m}\) with the property that for any vector \(X\in \mathfrak{m}\), there exists a vector \(a_X\in \mathfrak{h}\) such that \([a_X+X,AX] = 0\). Here, \(\mathfrak{g},\mathfrak{h}\) are the Lie algebras of \(G,H\) respectively, and \(\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}\) is the reductive decomposition with respect to an \(\mathrm{Ad}\)-invariant inner product in \(\mathfrak{g}\). In particular, \(\mathfrak{m}\cong_{\mathrm{iso}} T_pM\).
Several results are presented to simplify the study of the algebraic formulation of the problem. These are used to study the \(\operatorname{U}(n)\)-GO metrics on the complex Stiefel manifolds \(V_k\mathbb{C}^n\), proving that there exists a unique (up to scalar) one-parameter family \(A_t\) of \(\operatorname{U}(n)\)-GO metrics in \(V_k\mathbb{C}^n\) (see Theorem 1).


53C22 Geodesics in global differential geometry
53C30 Differential geometry of homogeneous manifolds
53C20 Global Riemannian geometry, including pinching


Zbl 1148.53038
Full Text: DOI arXiv


[1] Alekseevsky, D.; Arvanitoyeorgos, A., Riemannian flag manifolds with homogeneous geodesics, Trans. Amer. Math. Soc., 359, 3769-3789, (2007) · Zbl 1148.53038
[2] D. V. Alekseevsky, Y. G. Nikonorov, Compact Riemannian manifolds with homogeneous geodesics, SIGMA: Symmetry Integrability Geom. Methods Appl. 093 (2009), no. 5, 16 pp. · Zbl 1189.53047
[3] Arvanitoyeorgos, Andreas, Lie groups, 1-22, (2003), Providence, Rhode Island · Zbl 1045.53001
[4] Berestovskii, VN; Nikonorov, YG, On δ-homogeneous Riemannian manifolds, Differential Geom. Appl., 26, 514-535, (2008) · Zbl 1155.53022
[5] Berestovskii, VN; Nikonorov, YG, Clifford-Wolf homogeneous Riemannian manifolds, J. Differential Geom., 82, 467-500, (2009) · Zbl 1179.53043
[6] Berndt, J.; Kowalski, O.; Vanhecke, L., Geodesics in weakly symmetric spaces, Ann. Global Anal. Geom., 15, 153-156, (1997) · Zbl 0880.53044
[7] Berger, M., Les variétés Riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15, 179-246, (1961) · Zbl 0101.14201
[8] Calvaruso, G.; Marinosci, RA, Homogeneous geodesics of three-dimensional unimodular Lorentzian Lie groups, Mediterranean J. Math., 3, 467-481, (2006) · Zbl 1150.53010
[9] J. E. D’Atri, W. Ziller, Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups, Memoirs Amer. Math. Soc. 19 (1979), no. 215.
[10] Dušek, Z., Survey on homogeneous geodesics, Note di Mat., 1, 147-168, (2008) · Zbl 1198.53052
[11] Gordon, Carolyn S., Homogeneous Riemannian Manifolds Whose Geodesies Are Orbits, 155-174, (1996), Boston, MA · Zbl 0861.53052
[12] Э.Б\(.\) Винберг, А. Л. Онищик, Основы теории групп Ли, В. В. Горбацевич, А. Л. Онищик, Группы Ли преобразований, в томе Группы Ли и алгебры Ли-I, Итоги науки и техн., Совр. пробл. матем., фунд. направл., т. 20, ВИНИТИ, М., 1988, стр. 7-101, 103-244. Engl. transl.: A. L. Onishchik, E. B. Vinberg, Foundations of Lie theory, V. V. Gorbatsevich, A. L. Onishchik, Lie transformation groups, in: Lie Groups and Lie Algebra I, Encyclopaedia of Mathematical Sciences, Vol. 20, Springer-Verlag, Berlin, 2001, pp. 4-94 and pp. 99-229.
[13] Hsiang, WY; Su, JC, On the classification of transitive effective actions on Stiefel manifolds, Trans. Amer. Math. Soc., 130, 322-336, (1968) · Zbl 0199.27104
[14] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II, Wiley (Interscience), New York, 1969.
[15] Kowalski, O.; Vanhecke, L., A generalization of a theorem on naturally reductive homogeneous spaces, Proc. Amer. Math. Soc., 91, 433-435, (1984) · Zbl 0542.53029
[16] Kowalski, O.; Vanhecke, L., Riemannian manifolds with homogeneous geodesics, Un. Mat. Ital. B., 7, 189-246, (1991) · Zbl 0731.53046
[17] Milnor, J., Curvatures of left invariant metrics on Lie groups, Adv. in Math., 21, 293-329, (1976) · Zbl 0341.53030
[18] Nikonorov, YG, Geodesic orbit Riemannian metrics on spheres, Vladikavkaz. Mat. Zh., 15, 67-76, (2013) · Zbl 1293.53062
[19] H. Tamaru, Riemannian g.o. spaces fibered over irreducible symmetric spaces, Osaka J. Math. 36 (1999), 835-851. · Zbl 0963.53026
[20] J. A. Wolf, Harmonic Analysis on Commutative Spaces, Mathematical Surveys and Monographs, Vol. 142, American Mathematical Society, Providence, RI, 2007. · Zbl 1156.22010
[21] О. Я. Якимова, Слабо симметрические римановы мкногообразия, итеющие редуктивную группу изометрий, Матем. сб. 195 (2004), мер. 4, 143-160. Engl. transl.: O. S. Yakimova, Weakly symmetric Riemannian manifolds with reductive isometry group, Sb. Math. 195 (2004), no. 4, 599-614.
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