scholarly journals FANO MANIFOLDS, CONTACT STRUCTURES, AND QUATERNIONIC GEOMETRY

1995 ◽  
Vol 06 (03) ◽  
pp. 419-437 ◽  
Author(s):  
CLAUDE LEBRUN
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Contact Structure ◽  
Twistor Space ◽  
Positive Scalar Curvature ◽  
Einstein Metric ◽  
Contact Structures ◽  
Fano Manifolds ◽  
Quaternionic Geometry ◽  
Structure Formula

Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M4n, g). If Z also admits a second complex contact structure [Formula: see text], then Z=CP2n+1. As an application, we give several new characterizations of the Riemannian manifold HPn= Sp(n+1)/(Sp(n)×Sp(1)).

The Twistor Space of a 3-Sasakian Manifold

1997 ◽  
Vol 08 (01) ◽  
pp. 31-60 ◽  
Author(s):  
Charles P. Boyer ◽  
Krzysztof Galicki
Keyword(s):  
Scalar Curvature ◽  
Smooth Manifold ◽  
Twistor Space ◽  
Sasakian Manifold ◽  
Positive Scalar Curvature ◽  
Einstein Metric ◽  
Singular Case ◽  
Positive Scalar ◽  
Smooth Case ◽  
Quaternionic Kähler

Any compact 3-Sasakian manifold [Formula: see text] is a principal circle V-bundle over a compact complex orbifold [Formula: see text]. This orbifold has a contact Fano structure with a Kähler–Einstein metric of positive scalar curvature and it is the twistor space of a positive compact quaternionic Kähler orbifold [Formula: see text]. We show that many results known to hold when [Formula: see text] is a smooth manifold extend to this more general singular case. However, we construct infinite families of examples with [Formula: see text] which sharply differs from the smooth case, where there is only one such [Formula: see text].

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

FANO MANIFOLDS WITH GEOMETRIC STRUCTURES MODELED AFTER HOMOGENEOUS CONTACT MANIFOLDS

2000 ◽  
Vol 11 (09) ◽  
pp. 1203-1230 ◽  
Author(s):  
JAEHYUN HONG
Keyword(s):  
Symmetric Space ◽  
Scalar Curvature ◽  
Positive Scalar Curvature ◽  
Geometric Structures ◽  
Contact Manifolds ◽  
Fano Manifolds ◽  
Positive Scalar ◽  
Model Spaces ◽  
Quaternionic Kähler Manifold ◽  
Quaternionic Kähler

In this paper we present a study on geometric structures modeled after homogeneous contact manifolds and show that on Fano manifolds these geometric structures are locally isomorphic to the standard geometric structures on the model spaces. This conclusion is analogous to those of [13, 7]. We expect that this work will help prove the conjecture that a compact quaternionic Kähler manifold of positive scalar curvature is a quaternionic symmetric space [21].

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.

scholarly journals Contact Structures of Sasaki Type and Their Associated Moduli

2019 ◽  
Vol 6 (1) ◽  
pp. 1-30
Author(s):  
Charles P. Boyer
Keyword(s):  
Scalar Curvature ◽  
Present Article ◽  
Einstein Metrics ◽  
Contact Structure ◽  
Constant Scalar Curvature ◽  
Contact Geometry ◽  
Contact Structures ◽  
Sasakian Geometry ◽  
Sasakian Structures

Abstract This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.

scholarly journals Tight contact structures via admissible transverse surgery

2019 ◽  
Vol 28 (04) ◽  
pp. 1950032 ◽  
Author(s):  
J. Conway
Keyword(s):  
Contact Structure ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Tight Contact ◽  
Transverse Knots ◽  
Structure Formula ◽  
Heegaard Floer

We investigate the line between tight and overtwisted for surgeries on fibered transverse knots in contact 3-manifolds. When the contact structure [Formula: see text] is supported by the fibered knot [Formula: see text], we obtain a characterization of when negative surgeries result in a contact structure with nonvanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure [Formula: see text] supported by the mirror knot [Formula: see text]. We derive several corollaries about the existence of tight contact structures, L-space knots outside [Formula: see text], nonplanar contact structures, and nonplanar Legendrian knots.

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 $F$.

Positive scalar curvature metrics via end-periodic manifolds

2020 ◽  
Vol 5 (3) ◽  
pp. 639-676
Author(s):  
Michael Hallam ◽  
Varghese Mathai
Keyword(s):  
Scalar Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar
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