Existence of homogeneous metrics with prescribed Ricci curvature
Gould, Mark1 ; Pulemotov, Artem1
1 School of Mathematics and Physics The University of Queensland St Lucia, QLD 4072 (Australia)
Séminaire de théorie spectrale et géométrie, Tome 33 (2015-2016), pp. 47-54.

Consider a compact Lie group G and a closed subgroup H<G. Suppose T is a positive-definite G-invariant (0,2)-tensor field on the homogeneous space M=G/H. In this note, we state a sufficient condition for the existence of a G-invariant Riemannian metric on M whose Ricci curvature coincides with cT for some c>0. This condition is, in fact, necessary if the isotropy representation of M splits into two inequivalent irreducible summands. After stating the main result, we work out an example.

Publié le :
DOI : 10.5802/tsg.313
Gould, Mark 1 ; Pulemotov, Artem 1

1 School of Mathematics and Physics The University of Queensland St Lucia, QLD 4072 (Australia) 
@article{TSG_2015-2016__33__47_0,
     author = {Gould, Mark and Pulemotov, Artem},
     title = {Existence of homogeneous metrics with prescribed {Ricci} curvature},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {47--54},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {33},
     year = {2015-2016},
     doi = {10.5802/tsg.313},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/tsg.313/}
}
TY  - JOUR
AU  - Gould, Mark
AU  - Pulemotov, Artem
TI  - Existence of homogeneous metrics with prescribed Ricci curvature
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2015-2016
SP  - 47
EP  - 54
VL  - 33
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/tsg.313/
DO  - 10.5802/tsg.313
LA  - en
ID  - TSG_2015-2016__33__47_0
ER  - 
%0 Journal Article
%A Gould, Mark
%A Pulemotov, Artem
%T Existence of homogeneous metrics with prescribed Ricci curvature
%J Séminaire de théorie spectrale et géométrie
%D 2015-2016
%P 47-54
%V 33
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/tsg.313/
%R 10.5802/tsg.313
%G en
%F TSG_2015-2016__33__47_0
Gould, Mark; Pulemotov, Artem. Existence of homogeneous metrics with prescribed Ricci curvature. Séminaire de théorie spectrale et géométrie, Tome 33 (2015-2016), pp. 47-54. doi : 10.5802/tsg.313. http://www.numdam.org/articles/10.5802/tsg.313/

[1] Anastassiou, Stavros; Chrysikos, Ioannis The Ricci flow approach to homogeneous Einstein metrics on flag manifolds, J. Geom. Phys., Volume 61 (2011), pp. 1587-1600 | Zbl

[2] Arvanitoyeorgos, Andreas An introduction to Lie groups and the geometry of homogeneous spaces, Student Mathematical Library, 22, American Mathematical Society, 2003, xvi+141 pages | DOI | MR | Zbl

[3] Böhm, Christoph Homogeneous Einstein metrics and simplicial complexes, J. Differ. Geom., Volume 67 (2004) no. 1, pp. 79-165 http://projecteuclid.org/euclid.jdg/1099587730 | MR | Zbl

[4] Böhm, Christoph; Wang, McKenzie; Ziller, Wolfgang A variational approach for compact homogeneous Einstein manifolds, Geom. Funct. Anal., Volume 14 (2004) no. 4, pp. 681-733 | DOI | MR | Zbl

[5] Dickinson, William; Kerr, Megan M. The geometry of compact homogeneous spaces with two isotropy summands, Ann. Global Anal. Geom., Volume 34 (2008) no. 4, pp. 329-350 | DOI | MR | Zbl

[6] Gould, Mark; Zhang, Ruibin Unitary representations of basic classical Lie superalgebras, Lett. Math. Phys., Volume 20 (1990) no. 3, pp. 221-229 | DOI | MR | Zbl

[7] He, Chenxu Cohomogeneity one manifolds with a small family of invariant metrics, Geom. Dedicata, Volume 157 (2012), pp. 41-90 | DOI | MR | Zbl

[8] Kac, Victor G. Lie superalgebras, Adv. Math., Volume 26 (1977) no. 1, pp. 8-96 | DOI | MR | Zbl

[9] Nikonorov, Yurii Gennadyevich; Rodionov, Eugene D.; Slavskiĭ, Viktor V. Geometry of homogeneous Riemannian manifolds, J. Math. Sci. (N.Y.), Volume 146 (2007) no. 6, pp. 6313-6390 | DOI | MR | Zbl

[10] Pulemotov, Artem Metrics with prescribed Ricci curvature on homogeneous spaces, J. Geom. Phys., Volume 106 (2016), pp. 275-283 | DOI | MR | Zbl

[11] Pulemotov, Artem; Rubinstein, Yanir A. Ricci iteration on homogeneous spaces (to appear in Trans. Am. Math. Soc.)

[12] Wang, McKenzie; Ziller, Wolfgang Existence and nonexistence of homogeneous Einstein metrics, Invent. Math., Volume 84 (1986) no. 1, pp. 177-194 | DOI | MR | Zbl

Cité par Sources :