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Register a new userHomogeneous spaces, curvature and cohomology
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Martin Herrmann
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We give new counterexamples to a question of Karsten Grove, whether there are only finitely many rational homotopy types among simply connected manifolds satisfying the assumptions of Gromovs Betti number theorem. Our counterexamples are homogeneous Riemannian manifolds, in contrast to previous ones. They consist of two families in dimensions 13 and 22. Both families are nonnegatively curved with an additional upper curvature bound and differ already by the ring structure of their cohomology rings with complex coefficients. The 22-dimensional examples also admit almost nonnegative curvature operator with respect to homogeneous metrics.
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We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type $mathcal{B}$ surfaces which are not Type $mathcal{A}$ we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to $1$.
We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global kinematic formulas in the Euclidean case are formally identical, the local formulas in the hermitian case contain strictly more information than the global ones. Even if one is only interested in the flat hermitian case, i.e. $mathbb C^n$, it is necessary to study the family of all complex space forms, indexed by the holomorphic curvature $4lambda$, and the dependence of the formulas on the parameter $lambda$. We will also describe Wannerers recent proof of local additive kinematic formulas for unitarily invariant area measures.
We prove that a 2n-dimensional compact homogeneous nearly Kahler manifold with strictly positive sectional curvature is isometric to CP^{n}, equipped with the symmetric Fubini-Study metric or with the standard Sp(m)-homogeneous metric, n =2m-1, or to S^{6} as Riemannian manifold with constant sectional curvature. This is a positive answer for a revised version of a conjecture given by Gray.
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Chengs maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
In this paper, we give a complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $partial M$ having positive isotropic curvature. We prove that for a pair $(f, kappa)$ of a nontrivial smooth function $f: M to {Bbb R}$ and a nonnegative real number $kappa$, if $(M, g)$ having positive isotropic curvature satisfies $$ Ddf - (Delta f)g - f{rm Ric} = kappa g, $$ then $(M, g)$ is isometric to a geodesic ball in ${Bbb S}^n$ when $kappa >0$, and either $M$ isometric to ${Bbb S}^n_+$, or the product $I times {Bbb S}^{n-1}$, up to finite cover when $kappa =0$.
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