Geodesic
Homogeneous almost normal Riemannian manifolds
Matematičeskie trudy, Tome 16 (2013) no. 1, pp. 18-27.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this article, we introduce a newclass of compact homogeneous Riemannian manifolds $(M=G/H,\mu)$ almost normal with respect to a transitive Lie group $G$ of isometries for which by definition there exists a $G$-left-invariant and an $H$-right-invariant inner product $\nu$ such that the canonical projection $p\colon(G,\nu)\rightarrow(G/H,\mu)$ is a Riemannian submersion and the norm ${|\boldsymbol\cdot|}$ of the product $\nu$ is at least the bi-invariant Chebyshev normon $G$ defined by the space $(M,\mu)$. We prove the following results: Every homogeneous Riemannian manifold is almost normal homogeneous. Every homogeneous almost normal Riemannian manifold is naturally reductive and generalized normal homogeneous. For a homogeneous $G$-normal Riemannian manifold with simple Lie group $G$, the unit ball of the norm ${|\boldsymbol\cdot|}$ is a Löwner–John ellipsoid with respect to the unit ball of the Chebyshev norm; an analogous assertion holds for the restrictions of these norms to a Cartan subgroup of the Lie group $G$. Some unsolved problems are posed. 
@article{MT_2013_16_1_a1,
     author = {V. N. Berestovskiǐ},
     title = {Homogeneous almost normal {Riemannian} manifolds},
     journal = {Matemati\v{c}eskie trudy},
     pages = {18--27},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2013_16_1_a1/}
}
TY  - JOUR
AU  - V. N. Berestovskiǐ
TI  - Homogeneous almost normal Riemannian manifolds
JO  - Matematičeskie trudy
PY  - 2013
SP  - 18
EP  - 27
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2013_16_1_a1/
LA  - ru
ID  - MT_2013_16_1_a1
ER  - 
%0 Journal Article
%A V. N. Berestovskiǐ
%T Homogeneous almost normal Riemannian manifolds
%J Matematičeskie trudy
%D 2013
%P 18-27
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2013_16_1_a1/
%G ru
%F MT_2013_16_1_a1
V. N. Berestovskiǐ. Homogeneous almost normal Riemannian manifolds. Matematičeskie trudy, Tome 16 (2013) no. 1, pp. 18-27. http://geodesic.mathdoc.fr/item/MT_2013_16_1_a1/

[1] Adams Dzh. F., Lektsii po gruppam Li, Nauka, M., 1979 | MR | Zbl

[2] Berestovskii V. N., Nikonorov Yu. G., “Chebyshevskaya norma na algebre Li gruppy dvizhenii odnorodnogo finslerova mnogoobraziya”, Sovrem. matem. prilozh., 60, Algebra (2008), 98–121 | MR

[3] Berestovskii V. N., Nikonorov Yu. G., “O $\delta$-odnorodnykh rimanovykh mnogoobraziyakh. II”, Sib. matem. zhurn., 50:2 (2009), 267–278 | MR | Zbl

[4] Vinberg E. B., “Invariantnye normy v kompaktnykh algebrakh Li”, Funkts. analiz i ego pril., 2:2 (1968), 89–90 | MR | Zbl

[5] Yakimova O. S., “Slabo simmetricheskie rimanovy mnogoobraziya, imeyuschie reduktivnuyu gruppu izometrii”, Matem. sb., 195:4 (2004), 143–160 | DOI | MR | Zbl

[6] Ball K. M., “An elementary introduction to modern convex geometry”, Flavors of Geometry, Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge, 1997, 1–58 | MR | Zbl

[7] Berestovskii V. N., Guijarro L., “A metric characterization of Riemannian submersions”, Ann. Global Anal. Geom., 18:6 (2000), 577–588 | DOI | MR

[8] Berestovskii V. N., Nikitenko E. V., Nikonorov Yu. G., “Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic”, Differential Geom. Appl., 29:4 (2011), 533–546 | DOI | MR

[9] Berestovskii V. N., Nikonorov Yu. G., “On $\delta$-homogeneous Riemannian manifolds”, Differential Geom. Appl., 26:5 (2008), 514–535 | DOI | MR

[10] Berestovskii V. N., Nikonorov Yu. G., Generalized normal homogeneous Riemannian metrics on spheres and projective spaces, 29 Oct. 2012, 32 pp., arXiv: 1210.7727v1[math.DG]

[11] Berestovskii V. N., Plaut C., “Homogeneous spaces of curvature bounded below”, J. Geom. Anal., 9:2 (1999), 203–219 | DOI | MR | Zbl

[12] Berger M., “Les variétés riemanniennes homogeǹes normales simplement connexes à courbure strictement positive”, Ann. Scuola Norm. Sup. Pisa (3), 15:3 (1961), 179–246 | MR | Zbl

[13] John F., “Extremum problems with inequalities as subsidiary conditions”, Courant Anniversary Volume, Interscience Publishers, New York, 1948, 187–204 | MR

[14] Kostant B., “On differential geometry and homogeneous spaces. I”, Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 258–261 | DOI | MR | Zbl

[15] Kostant B., “On holonomy and homogeneous spaces”, Nagoya Math. J., 12:1 (1957), 31–54 | MR | Zbl

[16] Kowalski O., Vanhekke L., “Riemannian manifolds with homogeneous geodesics”, Boll. Un. Mat. Ital. B (7), 5:1 (1991), 189–246 | MR | Zbl

[17] Lewis A. S., “Group invariance and convex matrix analysis”, SIAM J. Matrix Anal. Appl., 17:4 (1996), 927–949 | DOI | MR | Zbl

[18] Nomizu K., “Invariant affine connections on homogeneous spaces”, Amer. J. Math., 76:1 (1954), 33–65 | DOI | MR | Zbl

[19] Selberg A., “Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series”, J. Indian Math. Soc. (N.S.), 20 (1956), 47–87 | MR | Zbl

[20] Szenthe J., “Sur la connexion naturelle à torsion nulle”, Acta. Sci. Math. (Szeged), 38:3–4 (1976), 383–398 | MR | Zbl

[21] Ziller W., “Weakly symmetric spaces”, Topics in Geometry, In Memory of Joseph D'Atri, Progr. Nonlinear Differential Equations Appl., 20, Birkhäuser Boston, Boston, MA, 1996, 355–368 | MR | Zbl