Constant curvature hypersurfaces of cylinders over space forms

TL;DR

This work completes the local classification of hypersurfaces with constant sectional curvature in products of two space forms by treating $\mathbb{R}^k\times \mathbb{S}^{n-k+1}$ and $\mathbb{R}^k\times \mathbb{H}^{n-k+1}$ for $2\le k\le n-1$. It establishes a nonexistence result for nonzero curvature in these ambient spaces (and for the base case $\mathbb{R}^2\times \mathbb{Q}^2_{\epsilon}$), and then provides a full description of flat ($c=0$) hypersurfaces, showing they arise as extrinsic products with Euclidean factors or as configurations built from horosphere/horocycle structures and flat surfaces. The analysis relies on the $R$-operator framework, Gauss-Codazzi equations, and splitting criteria, connecting to recent work that culminates in a complete classification of constant-curvature hypersurfaces in products of space forms. These results yield a comprehensive local picture and have implications for the geometry of submanifolds in product spaces.

Abstract

We classify the hypersurfaces of dimension n >= 3 with constant sectional curvature in the product spaces R^k x S^{n-k+1} and R^k x H^{n-k+1}, for 2 <= k <= n-1. Our results provide a complete description of these hypersurfaces and extend previous classifications of constant curvature submanifolds in product spaces of space forms.

Theorems & Definitions (14)

• Proposition 1
• Lemma 2
• proof
• Proposition 3
• Proposition 4
• Theorem 5
• Proposition 6
• Theorem 7
• Lemma 8
• Lemma 9