Torus actions on manifolds with positive intermediate Ricci curvature

Abstract

We study closed, simply connected manifolds with positive $2^{\mathrm{nd}}$-intermediate Ricci curvature and large symmetry rank. In odd dimensions, we show that they are spheres. In even dimensions other than $6$, we show that they must have positive Euler characteristic. Under stronger assumptions on the symmetry rank, we show that such even dimensional manifolds must have trivial odd degree integral cohomology, and if the second Betti number is no more than $1$, they are either spheres or complex projective spaces. In the process, we establish new tools for studying isometric actions on closed manifolds with positive $k^{\mathrm{th}}$-intermediate Ricci for values of $k\ge 2$. These tools include generalizations of the isotropy rank lemma, symmetry rank bound, and connectedness principle from the setting of positive sectional curvature.

arXiv commenttypo corrected. Published in J. Lond. Math. Soc

References in corpus (8)

• Focal Radius, Rigidity, and Lower Curvature Bounds
• Sectional and intermediate Ricci curvature lower bounds via Optimal Transport
• On the Hopf Conjecture with Symmetry
• Local symmetry rank bound for positive intermediate Ricci curvatures
• Positive curvature and torus symmetry in small dimensions, I -- Dimensions 10, 12, 14, and 16
• The flavour of intermediate Ricci and homotopy when studying submanifolds of symmetric spaces
• Infinite families of manifolds of positive $k^{\mathrm{th}}$-intermediate Ricci curvature with $k$ small
• Splitting of torus representations and applications in the Grove symmetry program

Cited by in corpus (3)

• Local symmetry rank bound for positive intermediate Ricci curvatures
• Infinite families of manifolds of positive $k^{\mathrm{th}}$-intermediate Ricci curvature with $k$ small
• Positive intermediate Ricci curvature with maximal symmetry rank