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AN EXOTIC T1S 4 WITH POSITIVE CURVATURE
"... Spaces of positive curvature play a special role in geometry. Although the class of manifolds with positive (sectional) curvature is expected to be relatively small, so far there are only a few known obstructions. Moreover, for closed simply connected manifolds these coincide with the known obstruct ..."
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Spaces of positive curvature play a special role in geometry. Although the class of manifolds with positive (sectional) curvature is expected to be relatively small, so far there are only a few known obstructions. Moreover, for closed simply connected manifolds these coincide with the known obstructions to nonnegative curvature which are: (1) the Betti number theorem of Gromov which asserts that the homology of a compact manifold with nonnegative sectional curvature has an a priori bound on the number of generators depending only on the dimension, and (2) a result of Lichnerowicz and Hitchin implying that a spin manifold with nontrivial Â genus or generalized a genus cannot admit a metric with non negative curvature. One way to gain further insight is to construct and analyze examples. This is quite difficult and has been achieved only a few times. Aside from the classical rank one symmetric spaces, i.e., the spheres and the projective spaces with their canonical metrics, and the recently proposed deformation of the socalled GromollMeyer sphere [PW2], examples were only found in the 60’s by Berger [Be], in the 70’s by Wallach [Wa] and by Aloff and Wallach [AW], in the 80’s by Eschenburg [E1, E2], and in the 90’s by Bazaikin [Ba]. The examples by Berger, Wallach and AloffWallach were shown, by Wallach in even dimensions [Wa] and by BerardBergery [BB] in odd dimensions,
The classification of simply connected biquotients of dimension at most 7 and 3 new examples of almost positively curved manifolds
, 2011
"... ..."
On the curvature of biquotients
, 2008
"... Abstract. As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of nonnegatively curved manifolds which contain either a point or an open dense set of points at which all 2planes have positive curvature. We study infinite families of ..."
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Abstract. As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of nonnegatively curved manifolds which contain either a point or an open dense set of points at which all 2planes have positive curvature. We study infinite families of biquotients defined by Eschenburg and Bazaikin from this viewpoint, together with torus quotients of S 3 × S 3. There exist many examples of (compact) manifolds with nonnegative curvature. All homogeneous spaces G/H and all biquotients G/U inherit nonnegative curvature from the biinvariant metric on G. Additionally, it is shown in [GZ] that all cohomogeneityone manifolds, namely manifolds admitting an isometric group action with onedimensional orbit space, with singular orbits of codimension ≤ 2 admit metrics with nonnegative curvature. On the other hand, the known examples with positive curvature are very sparse (see [Zi1] for a survey). Other than the rankone symmetric spaces there are isolated examples in dimensions 6,7,12,13 and 24 due to Wallach
DEVELOPMENTS AROUND POSITIVE SECTIONAL CURVATURE
, 902
"... Abstract. This is not in any way meant to be a complete survey on positive curvature. Rather it is a short essay on the fascinating changes in the landscape surrounding positive curvature. In particular, details and many results and references are not included, and things are not presented in chrono ..."
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Abstract. This is not in any way meant to be a complete survey on positive curvature. Rather it is a short essay on the fascinating changes in the landscape surrounding positive curvature. In particular, details and many results and references are not included, and things are not presented in chronological order. Spaces of positive curvature have always enjoyed a particular role in Riemannian geometry. Classically, this class of spaces form a natural and vast extension of spherical geometry, and in the last few decades their importance for the study of general manifolds with a lower curvature bound via Alexandrov geometry has become apparent. The importance of Alexandrov geometry to Riemannian geometry stems from the fact that there are several natural geometric operations that are closed in Alexandrov geometry but not in Riemanian geometry. These include taking Gromov Hausdorff limits, taking quotients by isometric group actions, and forming joins of positively curved spaces. In particular, limits (or quotients) of Riemanian manifolds with a lower (sectional) curvature bound are Alexandrov spaces, and only rarely Riemannian manifolds. Analyzing limits frequently involves blow ups leading to spaces with nonnegative curvature as, e.g., in Perelman’s work on the geometrization conjecture. Also the infinitesimal structure of an Alexandrov space is expressed via its “tangent spaces”, which are cones on positively curved spaces. Hence the collection of all compact positively curved spaces (up to scaling) agrees with the class of all possible socalled spaces of directions, in Alexandrov spaces. So spaces of positive curvature play the same role in Alexandrov geometry as round spheres do in Riemannian geometry. In addition to positively, and nonnegativey curved spaces, yet another class of spaces has emerged in the general context of convergence under a lower curvature bound, namely almost nonnegatively curved spaces. These are spaces allowing metrics with diameter say 1, and lower curvature bound arbitrarily close to 0. They are expected to play a role among spaces with a lower curvature bound, analogous to that almost flat spaces play for spaces with bounded curvature. In summary, the following classes of spaces play essential roles in the study of spaces with a lower curvature bound:
Classification of almost quarterpinched manifolds
 Proc. Amer. Math. Soc
"... Abstract. We show that if a simply connected manifold is almost quarter pinched then it is di¤eomorphic to a CROSS or sphere. 1. ..."
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Abstract. We show that if a simply connected manifold is almost quarter pinched then it is di¤eomorphic to a CROSS or sphere. 1.
Topological and differentiable sphere theorems for complete submanifolds
 Comm. Anal. Geom
"... ar ..."
RIEMANNIAN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE
"... Of special interest in the history of Riemannian geometry have been manifolds with positive sectional curvature. In these notes we want to give a survey of this subject and some recent developments. We start with some historical developments. 1. History and Obstructions It is fair to say that Rieman ..."
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Of special interest in the history of Riemannian geometry have been manifolds with positive sectional curvature. In these notes we want to give a survey of this subject and some recent developments. We start with some historical developments. 1. History and Obstructions It is fair to say that Riemannian geometry started with Gauss’s famous ”Disquisitiones generales ” from 1827 in which one finds a rigorous discussion of what we now call the Gauss curvature of a surface. Much has been written about the importance and influence of this paper, see in particular the article [Do] by P.Dombrowski for a careful discussion of its contents and influence during that time. Here we only make a few comments. Curvature of surfaces in 3space had been studied previously by a number of authors and was defined as the product of the principal curvatures. But Gauss was the first to make the surprising discovery that this curvature only depends on the intrinsic metric and not on the embedding. Here one finds for example the formula for the metric in the form ds2 = dr2 + f(r, θ) 2dθ2. Gauss showed that every metric on a surface has this form in ”normal ” coordinates and that it has curvature K = −frr/f. In fact one can take it as the definition of the Gauss curvature and proves Gauss’s famous ”Theorema Egregium” that the curvature is an intrinsic invariant and does not depend on the embedding in R3. He also proved a local version of what we nowadays call the GaussBonnet theorem (it is not clear what Bonnet’s contribution was to this result), which states that in a geodesic triangle ∆ with angles α, β, γ the Gauss curvature measures the angle ”defect”: Kdvol = α + β + γ − π Nowadays the Gauss Bonnet theorem also goes under its global formulation for a compact surface:
A POSITIVELY CURVED MANIFOLD HOMEOMORPHIC TO T1S^4
"... Spaces of positive curvature play a special role in geometry. Although the class of manifolds with positive (sectional) curvature is expected to be relatively small, so far there are only a few known obstructions. Moreover, for closed simply connected manifolds these coincide with the known obstruct ..."
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Spaces of positive curvature play a special role in geometry. Although the class of manifolds with positive (sectional) curvature is expected to be relatively small, so far there are only a few known obstructions. Moreover, for closed simply connected manifolds these coincide with the known obstructions to nonnegative curvature which are: (1) the Betti number theorem of Gromov which asserts that the homology of a compact manifold with nonnegative sectional curvature has an a priori bound on the number of generators depending only on the dimension, and (2) a result of Lichnerowicz and Hitchin implying that a spin manifold with nontrivial Â genus or generalized a genus cannot admit a metric with non negative curvature. One way to gain further insight is to construct and analyze examples. This is quite difficult and has been achieved only a few times. Aside from the classical rank one symmetric spaces, i.e., the spheres and the projective spaces with their canonical metrics, and the recently proposed deformation of the socalled GromollMeyer sphere [PW2], examples were only found in the 60’s by Berger [Be], in the 70’s by Wallach [Wa] and by Aloff and Wallach [AW], in the 80’s by Eschenburg [E1, E2], and in the 90’s by Bazaikin [Ba]. The examples by Berger, Wallach and AloffWallach were shown, by Wallach in even dimensions [Wa] and by BerardBergery [BB] in odd