The Classification of Simply Connected Manifolds of Positive Scalar Curvature

1980 ◽  
Vol 111 (3) ◽  
pp. 423 ◽  
Author(s):  
Mikhael Gromov ◽  
H. Blaine Lawson
Keyword(s):  
Scalar Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar ◽  
Simply Connected Manifolds ◽  
Simply Connected

scholarly journals The action of the mapping class group on metrics of positive scalar curvature

2021 ◽  
Author(s):  
Georg Frenck
Keyword(s):  
Scalar Curvature ◽  
Mapping Class Group ◽  
Class Group ◽  
Index Theorem ◽  
Positive Scalar Curvature ◽  
Mapping Class ◽  
High Dimensional ◽  
Positive Scalar ◽  
Simply Connected

AbstractWe present a rigidity theorem for the action of the mapping class group $$\pi _0({\mathrm{Diff}}(M))$$ π 0 ( Diff ( M ) ) on the space $$\mathcal {R}^+(M)$$ R + ( M ) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional $${\mathrm{Spin}}$$ Spin -manifolds.

scholarly journals On Moduli Spaces of Positive Scalar Curvature Metrics on Highly Connected Manifolds

2020 ◽  
Author(s):  
Michael Wiemeler
Keyword(s):  
Fundamental Group ◽  
Scalar Curvature ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Positive Scalar Curvature ◽  
Spin Manifold ◽  
Homotopy Groups ◽  
Positive Scalar ◽  
Simply Connected ◽  
Highly Connected Manifolds

Abstract Let $M$ be a simply connected spin manifold of dimension at least six, which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar curvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds that includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.

scholarly journals Einstein Four-Manifolds With Self-Dual Weyl Curvature of Nonnegative Determinant

2019 ◽  
Author(s):  
Peng Wu
Keyword(s):  
Scalar Curvature ◽  
Positive Scalar Curvature ◽  
The Self ◽  
Weyl Curvature ◽  
Positive Scalar ◽  
Four Manifolds ◽  
Simply Connected

Abstract We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.

scholarly journals Moduli Spaces of Metrics of Positive Scalar Curvature on Topological Spherical Space Forms

2020 ◽  
Vol 63 (4) ◽  
pp. 901-908
Author(s):  
Philipp Reiser
Keyword(s):  
Scalar Curvature ◽  
Moduli Spaces ◽  
Riemannian Metrics ◽  
Universal Cover ◽  
Positive Scalar Curvature ◽  
Spherical Space ◽  
Space Forms ◽  
Positive Scalar ◽  
Space Of Riemannian Metrics ◽  
Simply Connected

AbstractLet $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

Metrics of Positive Scalar Curvature on Spherical Space Forms

1996 ◽  
Vol 48 (1) ◽  
pp. 64-80 ◽  
Author(s):  
Boris Botvinnik ◽  
Peter B. Gilkey
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Positive Scalar Curvature ◽  
Spherical Space ◽  
Space Forms ◽  
Eta Invariant ◽  
Positive Scalar ◽  
Spherical Space Forms ◽  
Simply Connected ◽  
Spherical Space Form

AbstractWe use the eta invariant to show every non-simply connected spherical space form of dimension m ≥ 5 has a countable family of non bordant metrics of positive scalar curvature.

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