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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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Cited by 22 (4 self)
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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,
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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Cited by 5 (0 self)
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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
CONCAVITY AND RIGIDITY IN NONNEGATIVE CURVATURE
"... We show that for a manifold with nonnegative curvature one obtains a collection of concave functions, special cases of which are the concavity of the length of a Jacobi field in dimension 2, and the concavity of the volume in general. We use these functions to show that there are many cohomogenei ..."
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Cited by 4 (1 self)
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We show that for a manifold with nonnegative curvature one obtains a collection of concave functions, special cases of which are the concavity of the length of a Jacobi field in dimension 2, and the concavity of the volume in general. We use these functions to show that there are many cohomogeneity one manifolds which do not carry an analytic invariant metric with nonnegative curvature. This implies in particular, that one of the candidates in [GWZ] does not carry an invariant metric with positive curvature.
A new type of a positively curved manifold
"... 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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Cited by 2 (2 self)
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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 trivial 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 [PW], 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,