The flavour of intermediate Ricci and homotopy when studying submanifolds of symmetric spaces

Amann, Manuel; Quast, Peter; Zarei, Masoumeh

doi:10.48550/arxiv.2010.15742

Cited by 3 publications

(8 citation statements)

References 14 publications

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“…In particular, we list the values b(G/K ) for the irreducible symmetric spaces in Table 3 at the end of this article. Very recently, during the writing of this paper, Amann, Quast and Zarei obtained the same result independently, and used it to study the higher connectedness of symmetric spaces [3].…”

Section: Normal Homogeneous and Symmetric Spacesmentioning

confidence: 70%

“…Remark 1. 3 The Wilking metric of Ric 2 > 0 on S 3 × S 3 is left-invariant and right S 3 -invariant, see Theorem E. By Corollary 2.3 it induces metrics of Ric 2 > 0 on S 2 × S 2 and S 2 × S 3 , by taking the quotients of the left action of a maximal torus of S 3 ×S 3 , and of the right S 1 -action, respectively. The manifolds S 2 ×S 2 , S 2 ×S 3 , and S 3 × S 3 are the only simply connected ones we know of to admit metrics of Ric 2 > 0, besides those of sec > 0.…”

Section: Proof Of Theorem B For Anymentioning

confidence: 95%

“…By standard theory of root systems, one can calculate the dimensions of such totally geodesic submanifolds, thus deriving the value of b(G/K ). The proof in [3] is, in essence, equivalent to ours, the main difference being that we make use of the Dynkin diagrams instead of the explicit description of the roots.…”

Section: Normal Homogeneous and Symmetric Spacesmentioning

confidence: 99%

“…1.1 will follow. We refer the reader to the recent work [3] for an alternative discussion of the problem addressed in this section.…”

Section: Symmetric Spacesmentioning

confidence: 99%

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Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small

2022

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Positive $$k\mathrm{th}$$ k th -intermediate Ricci curvature on a Riemannian n-manifold, to be denoted by $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 , is a condition that interpolates between positive sectional and positive Ricci curvature (when $$k =1$$ k = 1 and $$k=n-1$$ k = n - 1 respectively). In this work, we produce many examples of manifolds of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 with k small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension $$n\ge 7$$ n ≥ 7 congruent to $$3\,{{\,\mathrm{mod}\,}}4$$ 3 mod 4 supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 for some $$k<n/2$$ k < n / 2 . We also prove that each dimension $$n\ge 4$$ n ≥ 4 congruent to 0 or $$1\,{{\,\mathrm{mod}\,}}4$$ 1 mod 4 supports closed manifolds which carry metrics of $${{\,\mathrm{Ric}\,}}_k>0$$ Ric k > 0 with $$k\le n/2$$ k ≤ n / 2 , but do not admit metrics of positive sectional curvature.

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Section: Normal Homogeneous and Symmetric Spacesmentioning

confidence: 70%

Section: Proof Of Theorem B For Anymentioning

confidence: 95%

Section: Normal Homogeneous and Symmetric Spacesmentioning

confidence: 99%

“…1.1 will follow. We refer the reader to the recent work [3] for an alternative discussion of the problem addressed in this section.…”

Section: Symmetric Spacesmentioning

confidence: 99%

See 2 more Smart Citations

Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small

2022

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“…There has recently been increased interest in studying manifolds with lower bounds on k th -intermediate Ricci curvature. For example, see [AQZ20], [Cha19], [DGM20], [GW], [KM18], [Rov00], and the references therein. In these works, one can see that a major motivation for studying manifolds with Ric k > 0 is that it provides restrictions on the geometry and topology of submanifolds.…”

mentioning

confidence: 99%

Torus actions on manifolds with positive intermediate Ricci curvature

Mouillé¹

2021

Preprint

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On an n-dimensional manifold, positive k th -intermediate Ricci curvature is a condition interpolating between positive sectional curvature (k = 1) and positive Ricci curvature (k = n − 1). In this article, we show that closed manifolds with positive 2 nd -intermediate Ricci curvature and large symmetry rank are either homeomorphic to spheres in odd dimensions or have positive Euler characteristic in even dimensions. In the process, we establish new tools for studying closed manifolds with torus symmetries and positive k th -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.1 Positive k th -intermediate Ricci curvature should not be confused with k-positive Ricci curvature; see, for example, [CW20] and the references therein.

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Torus actions on manifolds with positive intermediate Ricci curvature

Mouillé

2022

Journal of London Math Soc

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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.

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The flavour of intermediate Ricci and homotopy when studying submanifolds of symmetric spaces

Cited by 3 publications

References 14 publications

Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small

Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small

Torus actions on manifolds with positive intermediate Ricci curvature

Torus actions on manifolds with positive intermediate Ricci curvature

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