Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 068, 6 pages      arXiv:2005.00161      https://doi.org/10.3842/SIGMA.2020.068
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces

Yukai Sun a and Xianzhe Dai b
a)  School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200241, P.R. of China
b)  Department of Mathematics, UCSB, Santa Barbara CA 93106, USA

Received May 04, 2020, in final form July 22, 2020; Published online July 25, 2020

Abstract
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.

Key words: extremal/rigid metrics; Lie groups; homogeneous spaces; scalar curvature.

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References

  1. Cheeger J., Ebin D.G., Comparison theorems in Riemannian geometry, AMS Chelsea Publishing, Providence, RI, 2008.
  2. Dai X., Wang X., Wei G., On the variational stability of Kähler-Einstein metrics, Comm. Anal. Geom. 15 (2007), 669-693.
  3. Goette S., Scalar curvature estimates by parallel alternating torsion, Trans. Amer. Math. Soc. 363 (2011), 165-183, arXiv:0709.4586.
  4. Goette S., Semmelmann U., Scalar curvature estimates for compact symmetric spaces, Differential Geom. Appl. 16 (2002), 65-78, arXiv:math.DG/0010199.
  5. Gromov M., Positive curvature, macroscopic dimension, spectral gaps and higher signatures, in Functional Analysis on the Eve of the 21st Century, Vol. II (New Brunswick, NJ, 1993),Progr. Math., Vol. 132, Birkhäuser Boston, Boston, MA, 1996, 1-213.
  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.
  8. Kramer W., The scalar curvature on totally geodesic fiberings, Ann. Global Anal. Geom. 18 (2000), 589-600.
  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.
  11. Milnor J., Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293-329.
  12. Min-Oo M., Scalar curvature rigidity of certain symmetric spaces, in Geometry, Topology, and Dynamics (Montreal, PQ, 1995), CRM Proc. Lecture Notes, Vol. 15, Amer. Math. Soc., Providence, RI, 1998, 127-136.
  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.
  14. Schoen R., Yau S.T., On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), 159-183.
  15. Wang M.Y., Ziller W., Existence and nonexistence of homogeneous Einstein metrics, Invent. Math. 84 (1986), 177-194.

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