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Intermediate curvatures and highly connected manifolds
Diarmuid Crowley,David J. Wraith +1 more
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In this article, it was shown that all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature.Abstract:
We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive Ricci curvature is defined to mean that the sum of the two smallest eigenvalues of the Ricci tensor is positive at every point. This result is a counterpart to a previous result of the authors concerning the existence of positive Ricci curvature on highly connected manifolds in dimensions 4j-1 for j > 1, and in dimensions 4j+1 for j > 0 with torsion-free cohomology.read more
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Intermediate Ricci curvatures and Gromov's Betti number bound
Philipp Reiser,David J. Wraith +1 more
TL;DR: For Riemannian metrics with Ricci curvatures, the authors showed that Gromov's upper Betti number bound for the sectional curvature bounded below fails to hold for Ricci-k > 0.
Positive intermediate Ricci curvature with maximal symmetry rank
Lee Kennard,Lawrence Mouill'e +1 more
TL;DR: In this article , the second author proved upper bounds on the ranks of isometry groups of closed Riemannian manifolds with positive intermediate Ricci curvatures and established some topological rigidity results in the case of maximal symmetry rank and positive second intermediate curvature.
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Spaces of positive intermediate curvature metrics
Georg Frenck,Jan-Bernhard Kordaß +1 more
TL;DR: In this paper, it was shown that the spaces of Riemannian metrics with positive p-curvature and k-positive Ricci curvatures on a given high-dimensional Spin-manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.
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Infinite families of manifolds of positive $k^{\rm th}$-intermediate Ricci curvature with $k$ small
TL;DR: In this article, it was shown that every dimension of the Riemannian manifold congruent to the 3-mod 4 dimension supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of positive Ricci curvature.
Positive intermediate curvatures and Ricci flow
TL;DR: In this article , it was shown that Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvatures, and that there exists a homogeneous space of dimension $d = 8n-4$ with metrics whose Ricci tensor is not $(d-4)-positive.
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Groups of Homotopy Spheres: I
Michel Kervaire,John Milnor +1 more
TL;DR: In this article, the disjoint sum of two connected n-manifolds is obtained by removing a small n-cell from each, and then pasting together the resulting boundaries.
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Positive scalar curvature and the Dirac operator on complete riemannian manifolds
TL;DR: Theoreme d'indice relatif as mentioned in this paper generalises sur a complete variete complete and generalizes on varietes incompletes of dimension ≤ 7, which represent classes d'homologie non triviales dans des espaces de courbure non positive.
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Manifolds with positive curvature operators are space forms
Christoph Böhm,Burkhard Wilking +1 more
TL;DR: The Ricci flow was introduced by Hamilton in 1982 as discussed by the authors in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form.
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Manifolds with positive curvature operators are space forms
Christoph Boehm,Burkhard Wilking +1 more
TL;DR: In this article, it was shown that on compact manifolds, the normalized Ricci flow evolves metrics with positive curvature operators to limit metrics with constant curvature, which is a conjecture of Hamilton.