Ricci flow and curvature on the variety of flags on the two dimensional projective space over the complexes, quaternions and octonions

Cheung, Man-Wai; Wallach, Nolan

doi:10.1090/S0002-9939-2014-12241-6

Ricci flow and curvature on the variety of flags on the two dimensional projective space over the complexes, quaternions and octonions
HTML articles powered by AMS MathViewer

by Man-Wai Cheung and Nolan R. Wallach PDF

Proc. Amer. Math. Soc. 143 (2015), 369-378 Request permission

Abstract:

For homogeneous metrics on the spaces over the complexes, quaternions and octonions, it is shown that the Ricci flow can move a metric of strictly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with indefinite signature. A variant of the method of Böhm and Wilking is given, proving that one can flow a metric of positive sectional curvature to one with Ricci curvature of indefinite signature in the quaternionic and octonian cases. A proof is given that this cannot occur in the complex case.

References

Simon Aloff and Nolan R. Wallach, An infinite family of distinct $7$-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93–97. MR 370624, DOI 10.1090/S0002-9904-1975-13649-4
Christoph Böhm and Burkhard Wilking, Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature, Geom. Funct. Anal. 17 (2007), no. 3, 665–681. MR 2346271, DOI 10.1007/s00039-007-0617-8
Christoph Böhm and Burkhard Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math. (2) 167 (2008), no. 3, 1079–1097. MR 2415394, DOI 10.4007/annals.2008.167.1079
Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006. MR 2274812, DOI 10.1090/gsm/077
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. MR 862046
Dan Knopf, Positivity of Ricci curvature under the Kähler-Ricci flow, Commun. Contemp. Math. 8 (2006), no. 1, 123–133. MR 2208813, DOI 10.1142/S0219199706002052
Lei Ni, Ricci flow and nonnegativity of sectional curvature, Math. Res. Lett. 11 (2004), no. 5-6, 883–904. MR 2106247, DOI 10.4310/MRL.2004.v11.n6.a12
F. M. Valiev, Sharp estimates of sectional curvatures of homogeneous Riemannian metrics on Wallach spaces, Sibirsk. Mat. Zh. 20 (1979), no. 2, 248–262, 457 (Russian). MR 530489
Nolan R. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. (2) 96 (1972), 277–295. MR 307122, DOI 10.2307/1970789