Geodesic
Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group $G$ are extremal (in fact rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact the same result holds for a compact connected homogeneous manifold $G/H$ with $G$ compact connect and semi-simple.
Keywords: extremal/rigid metrics, Lie groups, homogeneous spaces, scalar curvature. 
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     author = {Yukai Sun and Xianzhe Dai},
     title = {Gromov {Rigidity} of {Bi-Invariant} {Metrics} on {Lie} {Groups} and {Homogeneous} {Spaces}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a67/}
}
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Yukai Sun; Xianzhe Dai. Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a67/

[1] Cheeger J., Ebin D. G., Comparison theorems in Riemannian geometry, AMS Chelsea Publishing, Providence, RI, 2008 | DOI | MR | Zbl

[2] Dai X., Wang X., Wei G., “On the variational stability of Kähler–Einstein metrics”, Comm. Anal. Geom., 15 (2007), 669–693 | DOI | MR | Zbl

[3] Goette S., “Scalar curvature estimates by parallel alternating torsion”, Trans. Amer. Math. Soc., 363 (2011), 165–183, arXiv: 0709.4586 | DOI | MR | Zbl

[4] Goette S., Semmelmann U., “Scalar curvature estimates for compact symmetric spaces”, Differential Geom. Appl., 16 (2002), 65–78, arXiv: math.DG/0010199 | DOI | MR | Zbl

[5] Gromov M., “Positive curvature, macroscopic dimension, spectral gaps and higher signatures”, Functional Analysis on the Eve of the 21st Century (New Brunswick, NJ, 1993), v. II, Progr. Math., 132, Birkhäuser Boston, Boston, MA, 1996, 1–213 | DOI | MR | Zbl

[6] Gromov M., Four lectures on scalar curvature, arXiv: 1908.10612

[7] Gromov M., Lawson Jr. H.B., “Spin and scalar curvature in the presence of a fundamental group. I”, Ann. of Math., 111 (1980), 209–230 | DOI | MR | Zbl

[8] Kramer W., “The scalar curvature on totally geodesic fiberings”, Ann. Global Anal. Geom., 18 (2000), 589–600 | DOI | MR | Zbl

[9] Listing M., Scalar curvature on compact symmetric spaces, arXiv: 1007.1832

[10] Llarull M., “Sharp estimates and the Dirac operator”, Math. Ann., 310 (1998), 55–71 | DOI | MR | Zbl

[11] Milnor J., “Curvatures of left invariant metrics on Lie groups”, Adv. Math., 21 (1976), 293–329 | DOI | MR | Zbl

[12] Min-Oo M., “Scalar curvature rigidity of certain symmetric spaces”, Geometry, Topology, and Dynamics (Montreal, PQ, 1995), CRM Proc. Lecture Notes, 15, Amer. Math. Soc., Providence, RI, 1998, 127–136 | DOI | MR | Zbl

[13] Schoen R., Yau S. T., “Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature”, Ann. of Math., 110 (1979), 127–142 | DOI | MR | Zbl

[14] Schoen R., Yau S. T., “On the structure of manifolds with positive scalar curvature”, Manuscripta Math., 28 (1979), 159–183 | DOI | MR | Zbl

[15] Wang M. Y., Ziller W., “Existence and nonexistence of homogeneous Einstein metrics”, Invent. Math., 84 (1986), 177–194 | DOI | MR | Zbl