A splitting theorem for open nonnegatively curved manifolds

We prove that an open nonnegatively curved manifold Mⁿ with soul S^k splits isometrically as S × ^2–k if (and only if) the holonomy group of the normal bundle NS is trivial.Supported by the Heinrich Hertz foundation     

A splitting theorem for manifolds of almost nonnegative Ricci curvature

Cheeger and Gromoll proved that a closed Riemannian manifold of nonnegative Ricci curvature is, up to a finite cover, diffeomorphic to a direct product of a simply connected manifold and a torus. In this paper, we extend this theorem to manifolds of almost nonnegative Ricci curvature.     

Volume and distance comparison theorems for sub-Riemannian manifolds

In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in [3] and its use to obtain sharp inequalities for solutions of the sub-Riemannian heat equation. As a consequence, we obtain a Gromov type precompactness theorem for the class of sub-Riemannian manifolds whose generalized Ricci curvature is bounded from below in the sense of [3].     

Intrinsic complements of equiregular sub-Riemannian manifolds

Under a nondegeneracy condition, we show that an equiregular sub-Riemannian manifold of step size \(r\) admits a canonical, \(V\) -rigid complement defined from the sub-Riemannian data that is preserved the by action of sub-Riemannian isometries. We explore how the existence of such a complement relates to results from the literature and study the step size 2 case in more detail.     

A splitting result for compact symplectic manifolds

We consider compact symplectic manifolds acted on effectively by a compact connected Lie group K in a Hamiltonian fashion. We prove that the squared moment map ∥μ∥² is constant if and only if K is semisimple and the manifold is K-equivariantly symplectomorphic to a product of a flag manifold and a compact symplectic manifold which is acted on trivially by K. In the almost-Kähler setting the symplectomorphism turns out to be an isometry.     

A splitting theorem for connected moravaK-theories

LetX be a connected, locally finite spectrum and letk(n) (n>-1) denote the (−1)-connected cover of then-th MoravaK-Theory associated to the primep.k(n) is aBP-module spectrum with π_*(k(n)) ≅ ℤ _p [υ _n ] where |v _n | = 2(p ⁿ -1). We prove the following splitting theorem: Thek(n) ^*-torsion ofk(n) ^* (X) is already annihilated byv _n ^e (e≥1) if and only ifk(n)ΛX is homotopy equivalent to a wedge of spectrak(n) and ^r k(n) (0≤r≤e-1) where ^r k(n) denotes ther-th Postnikov factor ofk(n). Moreover we investigate splitting conditions for ^r k(n)ΛX.     

A topological completeness theorem

We prove a topological completeness theorem for infinitary geometric theories with respect to sheaf models. The theorem extends a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected. Received: 27 September 1996 / Revised version: 15 January 1998     

A volume comparison theorem for Finsler manifolds

Let be a symmetric Finsler manifold, endowed with the Busemann volume form, and let be its unit disk bundle endowed with the canonical symplectic volume form. It is shown that , where is the volume of the unit disk in . Moreover, equality holds if and only if is Riemannian.

     

A Gauss-Bonnet-Chern theorem for complex Finsler manifolds

Using Chern’s method of transgression and the currents, we establish a Gauss-Bonnet-Chern theorem for general closed complex Finsler manifolds (M, F). This result extends the classical Gauss-Bonnet-Chern theorem for Hermitian manifolds. Furthermore, a simplified version of the Gauss-Bonnet-Chern theorem is obtained in the case of complex Berwald 1-manifolds.