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].

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 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)).

scholarly journals Enlargeable length-structure and scalar curvatures

2021 ◽  
Author(s):  
Jialong Deng
Keyword(s):  
Scalar Curvature ◽  
Smooth Manifold ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Positive Scalar Curvature ◽  
Smooth Manifolds ◽  
Connected Sum ◽  
Positive Scalar ◽  
Topological Manifolds ◽  
Arbitrary Dimensions

AbstractWe define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

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

scholarly journals Complete hypersurfaces in cylinders

1989 ◽  
Vol 46 (2) ◽  
pp. 313-318
Author(s):  
Thomas Hasanis
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar ◽  
Coordinate Functions ◽  
Bounded Below ◽  
Complete Hypersurfaces

AbstractWe consider the extent of certain complete hypersurfaces of Euclidean space. We prove that every complete hypersurface in En+1 with sectional curvature bounded below and non-positive scalar curvature has at least (n − 1) unbounded coordinate functions.

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