[Paper Review] Nonpositively curved manifolds containing a prescribed nonpositively curved hypersurface

T. Tâm Nguyễn Phan|arXiv (Cornell University)|Jun 1, 2012

Geometric Analysis and Curvature Flows2 references3 citations

TL;DR

This paper constructs a closed, nonpositively curved (n+1)-dimensional manifold of geometric rank one that contains any given closed, nonpositively curved n-manifold as a totally geodesic submanifold. Using pinched smooth hyperbolization and warped product metrics with convex warping functions, the authors embed the original manifold isometrically while preserving nonpositive curvature and ensuring the resulting manifold is not a product.

ABSTRACT

We use pinched smooth hyperbolization to show that every closed, nonpositively curved $n$-dimensional manifold $M$ can be embedded as a totally geodesic submanifold of a closed, nonpositively curved $(n+1)$-dimensional manifold $\hat{M}$ of geometric rank one.

Motivation & Objective

• To answer Ralf Spatzier's question on whether a closed, nonpositively curved n-manifold can be isometrically embedded as a totally geodesic submanifold in a higher-dimensional, rank-one, nonpositively curved manifold.
• To construct a closed (n+1)-dimensional manifold with sectional curvature ≤ 0 and geometric rank one that contains the original n-manifold as a totally geodesic submanifold.
• To ensure the resulting manifold is not a product, by achieving geometric rank one through the construction.
• To preserve nonpositive curvature during the metric smoothing process at the gluing locus.
• To establish the construction under the condition that the Whitehead group of the original manifold is trivial, which holds due to its nonpositive curvature.

Proposed method

• Use a smooth triangulation of the original manifold M and extend it to M×[0,1], then cone off the boundary to form a simplicial complex X with a single singular point.
• Apply strict hyperbolization to X to obtain h(X), a manifold with one singularity, ensuring hyperbolization pieces have sufficient width for pinched smooth hyperbolization.
• Remove the singular point to obtain W = h(X)\{h(*)}, a noncompact manifold with two ends homeomorphic to M×(0,∞).
• Equip W with a metric g of negative curvature such that each end is isometric to M×(a,∞) with metric dt² + e⁻²ᵗgₘ, for a < -1.
• Truncate W at t=0 and glue the two boundary components to form a closed manifold Ŵ with a metric g̅ that is a warped product dt² + e⁻²|ᵗ|gₘ on (-1,1)×M.
• Smooth the warping function e⁻²|ᵗ| around t=0 using a convex, smooth, even function φ(t) that matches e⁻²|ᵗ| outside a neighborhood of 0 and attains a minimum at t=0, preserving nonpositive curvature via the Bishop-O'Neill formula.

Experimental results

Research questions

• RQ1 Can every closed, nonpositively curved n-manifold be isometrically embedded as a totally geodesic submanifold in a closed, nonpositively curved (n+1)-manifold of geometric rank one?
• RQ2 Is it possible to construct such an embedding while ensuring the resulting manifold is not a product?
• RQ3 Can the metric on the resulting manifold be smoothed without violating nonpositive curvature?
• RQ4 What conditions on the original manifold are necessary for the construction to work, particularly regarding the Whitehead group?
• RQ5 How can pinched smooth hyperbolization be used to construct a rank-one, nonpositively curved manifold containing a given hypersurface?

Key findings

• The construction yields a closed, (n+1)-dimensional Riemannian manifold Ŵ with sectional curvature ≤ 0 and geometric rank one.
• The original manifold M embeds isometrically into Ŵ as a totally geodesic submanifold via the cross-section at t=0.
• The resulting metric on Ŵ is smooth and has nonpositive curvature due to the use of a convex, smooth, even warping function φ(t).
• The construction is valid when the Whitehead group of M is trivial, which is guaranteed by Farrell–Jones results for nonpositively curved manifolds of dimension >4.
• The metric on each end of the intermediate manifold W is isometric to M×(a,∞) with metric dt² + e⁻²ᵗgₘ, ensuring compatibility with the final warped product structure.
• The method ensures that the final manifold Ŵ is not a product, as confirmed by its geometric rank one, which follows from the curvature and topology of the construction.