Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1802.07836

Open manifolds with non-homeomorphic positively curved souls

David González-Álvaro·Marcus Zibrowius
View original source ↗Read source document ↗Export metadata ↓

Abstract

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class of spaces that includes all Eschenburg spaces.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

David González-Álvaro, Marcus Zibrowius. 2019-05-15. Open manifolds with non-homeomorphic positively curved souls. https://doi.org/10.1017/s0305004119000227

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Canonical metrics on holomorphic fibre bundles

In this article we completely describe the existence of canonical metrics, known as optimal symplectic connections, on isotrivial Kähler fibrations. In this setting an optimal symplectic connection is induced from a Hermite--Einstein connection on the holomorphic principal bundle of relative automorphisms, and the Hitchin--Kobayashi correspondence asserts the existence of such a connection precisely when the principal bundle is polystable. Combined with results of Dervan and Sektnan this generates many new examples of cscK metrics on the total space of holomorphic fibre bundles. Our results indicate that in general the optimal symplectic connection equation should be viewed as a generalisation of the Hermite--Einstein equation to holomorphic fibrations where the complex structure of the fibres varies.

math.DG↗
arXiv2026

Higher Fundamental Forms and Warped Product Hypersurfaces

Warped products are one of the simplest families of Riemannian manifolds that can have non-trivial geometries. In this article, we characterize the geometry of hypersurface embeddings arising from warped product manifolds using the language of higher (Riemannian) fundamental forms. In a similar vein, we also study the geometry of conformal manifolds with embedded hypersurfaces that admits a trivialization of the conformal metric to a product metric, with base manifold given by the embedded hypersurface. We show that the higher conformal fundamental forms play a critical role in their characterization.

math.DG↗
arXiv2026

Pansu pullback and spectral complexes

In this paper, we prove the commutativity between the Pansu pullback of a smooth contact map between Carnot groups and the differentials appearing in the spectral complexes. As a direct application, we also present a way of "lifting" a Pansu derivative (viewed as a Lie algebra homomorphism) from Carnot groups to their central extensions.

math.DG↗