Torus actions on manifolds with positive intermediate Ricci curvature

Abstract

We study closed, simply connected manifolds with positive 2 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 𝑘th-intermediate Ricci for values of 𝑘 ⩾ 2. These tools include generalizations of the isotropy rank lemma, symmetry rank bound, and connectedness principle from the setting of positive sectional curvature.

Symmetries of Spaces with Lower Curvature Bounds

“…A natural starting point to consider symmetries in the presence of positive intermediate Ricci curvature is to consider those with Ric 2 > 0. Mouillé [18] in recent work has been able to extend the Maximal Symmetry Rank Theorem to closed, 𝑛-manifolds with Ric 2 > 0, showing that the symmetry rank of such manifolds is the same as for the positive curvature case, and moreover obtains a similar classification result.…”

“…there are also metrics of positive Ric 𝑘 ′ with 𝑘 ′ < 𝑘 on such products. In particular, in Example 2.3 of Mouillé [18], he shows that one can put Ric 2 > 0 metrics on 𝑀 6 = 𝑆 3 × 𝑆 3 with 𝑇 3 symmetry, as well as Ric 2 > 0 on quotients by free torus actions of 𝑀 6 . These results are consistent with the almost maximal symmetry rank classification results mentioned earlier and suggest a connection between closed, simply-connected manifolds of nonnegative curvature with large symmetry rank and 𝑅𝑖𝑐 𝑘 > 0 for small 𝑘.…”

Positive intermediate curvatures and Ricci flow

“…In particular, they have shown that Gromov's Betti number bound fails in the case of Ric [ d 2 ]+2 > 0 in any dimension d ≥ 5. Finally, we mention that the bound k ≤ d 2 also appears in several structural results of manifolds of Ric k > 0, see [GW,KM,Mo22,Xi97]. We refer to [Mo] for a collection of publications and preprints concerning the curvature conditions Ric k > 0.…”

On the dynamics of positively curved metrics on SU(3)/T 2 under the homogeneous Ricci flow

“…Here, we follow the good description in [19]. For the forthcoming definitions, there is no widely used notation/terminology.…”