scholarly journals Torus actions on manifolds of positive sectional curvature

2003 ◽  
Vol 191 (2) ◽  
pp. 259-297 ◽  
Author(s):  
Burkhard Wilking
Keyword(s):  
Sectional Curvature ◽  
Positive Sectional Curvature ◽  
Torus Actions

scholarly journals Positively curved 6-manifolds with simple symmetry groups

2002 ◽  
Vol 74 (4) ◽  
pp. 589-597 ◽  
Author(s):  
FUQUAN FANG
Keyword(s):  
Sectional Curvature ◽  
Isometry Group ◽  
Symmetry Groups ◽  
Positive Sectional Curvature ◽  
Identity Component ◽  
Simply Connected ◽  
Lie Subgroup ◽  
Positively Curved

Let M be a simply connected compact 6-manifold of positive sectional curvature. If the identity component of the isometry group contains a simple Lie subgroup, we prove that M is diffeomorphic to one of the five manifolds listed in Theorem A.

scholarly journals Bounding Lagrangian widths via geodesic paths

2014 ◽  
Vol 150 (12) ◽  
pp. 2143-2183 ◽  
Author(s):  
Matthew Strom Borman ◽  
Mark McLean
Keyword(s):  
Sectional Curvature ◽  
New Method ◽  
Finite Width ◽  
Energy Capacity ◽  
Positive Sectional Curvature ◽  
Floer Cohomology ◽  
The Real ◽  
Ambient Manifold ◽  
Open Map ◽  
Geodesic Paths

AbstractThe width of a Lagrangian is the largest capacity of a ball that can be symplectically embedded into the ambient manifold such that the ball intersects the Lagrangian exactly along the real part of the ball. Due to Dimitroglou Rizell, finite width is an obstruction to a Lagrangian admitting an exact Lagrangian cap in the sense of Eliashberg–Murphy. In this paper we introduce a new method for bounding the width of a Lagrangian$Q$by considering the Lagrangian Floer cohomology of an auxiliary Lagrangian$L$with respect to a Hamiltonian whose chords correspond to geodesic paths in$Q$. This is formalized as a wrapped version of the Floer–Hofer–Wysocki capacity and we establish an associated energy–capacity inequality with the help of a closed–open map. For any orientable Lagrangian$Q$admitting a metric of non-positive sectional curvature in a Liouville manifold, we show the width of$Q$is bounded above by four times its displacement energy.

scholarly journals Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

2018 ◽  
Vol 2020 (5) ◽  
pp. 1346-1365 ◽  
Author(s):  
Jason DeVito ◽  
Ezra Nance
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Lie Group ◽  
Positive Curvature ◽  
Positive Sectional Curvature ◽  
Compact Symmetric Space ◽  
Rank 2 ◽  
Set Of Points ◽  
Positively Curved

Abstract A Riemannian manifold is said to be almost positively curved if the set of points for which all two-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented two-planes in $\mathbb{R}^{7}$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group $\mathbf{G}_{2}$ and the octonions, so do not obviously generalize to any other Grassmannians.

scholarly journals Rigidity of horospherical foliations

1989 ◽  
Vol 9 (1) ◽  
pp. 191-205 ◽  
Author(s):  
Dave Witte
Keyword(s):  
Sectional Curvature ◽  
Finite Volume ◽  
Symmetric Spaces ◽  
Positive Sectional Curvature ◽  
Algebraic Form ◽  
Locally Symmetric Spaces

AbstractM. Ratner's theorem on the rigidity of horocycle flows is extended to the rigidity of horospherical foliations on bundles over finite-volume locally-symmetric spaces of non-positive sectional curvature, and to other foliations of the same algebraic form.

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