Spaces of nonnegatively curved surfaces

Банах, Тарас; Belegradek, Igor

doi:10.2969/jmsj/07027344

Cited by 5 publications

(5 citation statements)

References 21 publications

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“…Especially in recent years there has been intensive activity and substantial further progress on these issues, compare, for example, [2,3,[6][7][8][9][10][11][13][14][15][16][17][18][20][21][22][25][26][27][28][29][30][31][33][34][35][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]58,60,61,[65][66][67][69]…”

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confidence: 99%

“…Moreover, in [65] an analogous result is shown for spaces of non-negative Ricci curvature metrics. Also, there are results about topological properties of the space of non-negatively curved metrics on two-spheres and real projective planes and on the Gromov-Hausdorff metric on the set of isometry classes of non-negatively curved two-spheres, compare [3,6]. (For the results which are known in the non-compact case please see the paragraph following Remark 1.7 below.…”

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confidence: 99%

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On the topology of moduli spaces of non-negatively curved Riemannian metrics

Tuschmann

Wiemeler

2021

Math. Ann.

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We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

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confidence: 99%

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confidence: 99%

On the topology of moduli spaces of non-negatively curved Riemannian metrics

Tuschmann

Wiemeler

2021

Math. Ann.

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“…In recent times there has indeed been much activity and progress on these issues, compare, for example, [BB18], [Bel17], [Bel18], [BFK17], [BKS11], [BKS15], [BH15], [BERW17], [BG95], [BG96], [BHSW10], [BWW17], [BFJ16], [Bus16], [Car88], [CH16], [Che04], [CM12], [CS13], [CSS17], [Des17], [DGA19], [DKT18], [FGKO17], [FG16], [FO09], [FO10a], [FO10b], [FS17], [Gaj87], [Goo17], [GL83], [HSS14], [Hit74], [KPT05], [KS93], [LM89], [Loh95], [RS01], [Rub01], [SW19], [TW15], [Tus16], [Wal11], [Wal13], [Wal14], [Wra11], [Wra16],…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, in [SW19] an analogous result is shown for spaces of nonnegative Ricci curvature metrics. Also, there are results about topological properties of the space of non-negatively curved metrics on two-spheres and real projective planes and on the Gromov-Hausdorff metric on the set of isometry classes of non-negatively curved two-spheres, compare [BB18], [Bel17]. (For the results which are known in the non-compact case please see the paragraph following Remark 1.7 below.…”

Section: Introductionmentioning

confidence: 99%

On the topology of moduli spaces of non-negatively curved Riemannian metrics

Tuschmann,

Wiemeler

2017

Preprint

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We study spaces and moduli spaces of Riemannian metrics with nonnegative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics. * W.T. acknowledges support from DFG Priority Program SPP 2026 'Geometry at Infinity'. † M.W. acknowledges support by DFG Grant HA 3160/11-1 and SFB 878.

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“…Such curvature conditions include positive scalar curvature, positive Ricci curvature, and non-negative sectional curvature. For some recent results concerning closed manifolds, see for example [8], [7], [20], [13], [14], [12], [4], [35], [36], [39], [16], [33], and the book [34].…”

Section: Introductionmentioning

confidence: 99%

Non-negative versus positive scalar curvature

Schick,

Wraith

2016

Preprint

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In this note, we look at the difference, or rather the absence of a difference, between the space of metrics of positive scalar curvature and metrics of non-negative scalar curvature. The main tool to analyze the former on a spin manifold is the spectral theory of the Dirac operator and refinements thereof. This can be used, for example, to distinguish between path components in the space of positive scalar curvature metrics. Despite the fact that non-negative scalar curvature a priori does not have the same spectral implications as positive scalar curvature, we show that all invariants based on the Dirac operator extend over the bigger space. Under mild conditions we show that the inclusion of the space of metrics of positive scalar curvature into that of non-negative scalar curvature is a weak homotopy equivalence.

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Spaces of nonnegatively curved surfaces

Cited by 5 publications

References 21 publications

On the topology of moduli spaces of non-negatively curved Riemannian metrics

On the topology of moduli spaces of non-negatively curved Riemannian metrics

On the topology of moduli spaces of non-negatively curved Riemannian metrics

Non-negative versus positive scalar curvature

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