scholarly journals Volume and macroscopic scalar curvature

2021 ◽  
Author(s):  
Sabine Braun ◽  
Roman Sauer
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Upper Bound ◽  
Betti Numbers ◽  
Universal Cover ◽  
Positive Scalar Curvature ◽  
Simplicial Volume ◽  
Positive Scalar ◽  
Curvature Bound

AbstractWe prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of $$\ell ^2$$ ℓ 2 -Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover.

Sewing Riemannian Manifolds with Positive Scalar Curvature

2017 ◽  
Vol 28 (4) ◽  
pp. 3553-3602 ◽  
Author(s):  
J. Basilio ◽  
J. Dodziuk ◽  
C. Sormani
Keyword(s):  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Positive Scalar Curvature ◽  
Positive Scalar

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.

scholarly journals Width, Largeness and Index Theory

2020 ◽  
Author(s):  
Rudolf Zeidler ◽  
◽  
◽  
Keyword(s):  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Upper Bound ◽  
Index Theory ◽  
Positive Scalar Curvature ◽  
Point Of View ◽  
Open Problems ◽  
Spin Manifolds ◽  
Recent Developments ◽  
Theoretic Point

In this note, we review some recent developments related to metric aspects of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands M×[−1,1], and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on M×R. We show that there is a more general geometric statement underlying both of them implying a quantitative negative upper bound on the infimum of the scalar curvature of a complete metric on M×R if the scalar curvature is positive in some neighborhood. We study (A hat-)iso-enlargeable spin manifolds and related notions of width for Riemannian manifolds from an index-theoretic point of view. Finally, we list some open problems arising in the interplay between index theory, largeness properties and width.

scholarly journals PARTIAL CLASSIFICATION RESULTS FOR POSITIVE QUATERNION KÄHLER MANIFOLDS

2012 ◽  
Vol 23 (02) ◽  
pp. 1250038
Author(s):  
MANUEL AMANN
Keyword(s):  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Positive Scalar Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
The Real ◽  
Positive Scalar ◽  
Partial Classification ◽  
Almost All

Positive Quaternion Kähler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. We prove this conjecture in dimension 20 under additional assumptions and we provide recognition theorems for the real Grassmannian [Formula: see text] in almost all dimensions.

scholarly journals Scalar curvature and the multiconformal class of a direct product Riemannian manifold

2021 ◽  
Author(s):  
Saskia Roos ◽  
Nobuhiko Otoba
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Direct Product ◽  
Warped Product ◽  
Riemannian Metrics ◽  
Positive Function ◽  
Positive Scalar Curvature ◽  
Conformal Class ◽  
Positive Scalar ◽  
Technical Assumption

AbstractFor a closed, connected direct product Riemannian manifold $$(M, g)=(M_1, g_1) \times \cdots \times (M_l, g_l)$$ ( M , g ) = ( M 1 , g 1 ) × ⋯ × ( M l , g l ) , we define its multiconformal class $$ [\![ g ]\!]$$ [ [ g ] ] as the totality $$\{f_1^2g_1\oplus \cdots \oplus f_l^2g_l\}$$ { f 1 2 g 1 ⊕ ⋯ ⊕ f l 2 g l } of all Riemannian metrics obtained from multiplying the metric $$g_i$$ g i of each factor $$M_i$$ M i by a positive function $$f_i$$ f i on the total space M. A multiconformal class $$ [\![ g ]\!]$$ [ [ g ] ] contains not only all warped product type deformations of g but also the whole conformal class $$[\tilde{g}]$$ [ g ~ ] of every $$\tilde{g}\in [\![ g ]\!]$$ g ~ ∈ [ [ g ] ] . In this article, we prove that $$ [\![ g ]\!]$$ [ [ g ] ] contains a metric of positive scalar curvature if and only if the conformal class of some factor $$(M_i, g_i)$$ ( M i , g i ) does, under the technical assumption $$\dim M_i\ge 2$$ dim M i ≥ 2 . We also show that, even in the case where every factor $$(M_i, g_i)$$ ( M i , g i ) has positive scalar curvature, $$ [\![ g ]\!]$$ [ [ g ] ] contains a metric of scalar curvature constantly equal to $$-1$$ - 1 and with arbitrarily large volume, provided $$l\ge 2$$ l ≥ 2 and $$\dim M\ge 3$$ dim M ≥ 3 .

scholarly journals On compact Einstein doubly warped product manifolds

2018 ◽  
Vol 49 (4) ◽  
pp. 267-275 ◽  
Author(s):  
Punam Gupta
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Warped Product ◽  
Positive Scalar Curvature ◽  
Positive Scalar

In this paper, the non-existence of connected, compact Einstein doubly warped product semi-Riemannian manifold with non-positive scalar curvature is proved. It is also shown that there does not exist non-trivial connected Einstein doubly warped product semi-Riemannian manifold with compact base $B$ or fibre $F$.

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