scholarly journals Rigidity of lattices of non-positive curvature

1983 ◽  
Vol 3 (1) ◽  
pp. 47-85 ◽  
Author(s):  
P. Eberlein
Keyword(s):  
Symmetric Space ◽  
Sectional Curvature ◽  
Positive Curvature ◽  
Fundamental Groups ◽  
Positive Sectional Curvature ◽  
Constant Multiple ◽  
De Rham

AbstractLet M, M* denote compact, connected manifolds of non-positive sectional curvature whose fundamental groups are isomorphic and whose Euclidean de Rham factors are trivial. We prove that: if M is a compact irreducible quotient of a reducible symmetric space H, then M and M* are isometric up to a constant multiple of the metric; and that the number and dimensions of the local de Rham factors are the same for M and M*. Gromov has independently proved the first result in the more general case that M is locally symmetric and globally irreducible with rank at least two.

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 On the growth of fundamental groups of nonpositive curvature manifolds

1996 ◽  
Vol 54 (3) ◽  
pp. 483-487 ◽  
Author(s):  
Yi-Hu Yang
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Sectional Curvature ◽  
Exponential Growth ◽  
Compact Riemannian Manifold ◽  
Nonpositive Curvature ◽  
Fundamental Groups ◽  
Positive Sectional Curvature ◽  
Classic Result ◽  
Negative Sectional Curvature

Milnor's classic result that the fundamental group of a compact Riemannian manifold of negative sectional curvature has exponential growth is generalised to the case of negative Ricci curvature and non-positive sectional curvature.

Positive curvature and symmetry in small dimensions

2019 ◽  
Vol 22 (06) ◽  
pp. 1950053 ◽  
Author(s):  
Manuel Amann ◽  
Lee Kennard
Keyword(s):  
Sectional Curvature ◽  
Euler Characteristic ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Elliptic Genus ◽  
Euler Characteristics ◽  
Positive Sectional Curvature ◽  
Related Family ◽  
Diffeomorphism Type

Extending existing work in small dimensions, Dessai computed the Euler characteristic, signature, and elliptic genus for [Formula: see text]-manifolds of positive sectional curvature in the presence of torus symmetry. He also computes the diffeomorphism type by restricting his results to classes of manifolds known to admit non-negative curvature, such as biquotients. The first part of this paper extends Dessai’s calculations to even dimensions up to [Formula: see text]. In particular, we obtain a first characterization of the Cayley plane in such a setting. The second part studies a closely related family of manifolds called positively elliptic manifolds, and we prove a conjecture of Halperin in this context for dimensions up to [Formula: see text] or Euler characteristics up to [Formula: see text].

scholarly journals Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

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.

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