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Dihedral rigidity for submanifolds of warped product manifolds

Jinmin Wang, Zhizhang Xie

TL;DR

This paper proves Gromov's dihedral extremality and rigidity conjectures for a wide class of codimension-zero submanifolds with polyhedral boundary inside warped product spaces, including cases where the submanifold faces are neither orthogonal nor parallel to the radial direction. It develops a spinorial framework using a warped-product-derived potential in a Dirac operator and leverages index theory for polyhedral boundaries to obtain sharp curvature and dihedral-angle equalities under nonzero degree maps. The results cover radially convex domains in warped products and extend to hyperbolic manifolds (notably parabolic prisms in $\mathbb{H}^n$) and certain direct-product spaces, yielding rigidity statements that force local isometries or product decompositions when curvature, dihedral, and boundary data saturate the comparisons. Overall, the work broadens the scope of dihedral rigidity from nonnegative-curvature leaves to hyperbolic settings and product geometries, with implications for scalar curvature, boundary geometry, and geometric-topological rigidity phenomena. The methods unify index-theoretic techniques with a spinorial approach to warped-product metrics and provide a robust framework for Stoker-type and related rigidity results in polyhedral geometry.

Abstract

In this paper, we prove a dihedral extremality and rigidity theorem for a large class of codimension zero submanifolds with polyhedral boundary in warped product manifolds. We remark that the spaces considered in this paper are not necessarily warped product manifolds themselves. In particular, the results of this paper are applicable to submanifolds (of warped product manifolds) with faces that are neither orthogonal nor parallel to the radial direction of the warped product metric. Generally speaking, the dihedral rigidity results require the leaf of the underlying warped space to have positive Ricci curvature and the warping function to be strictly log-concave. Nevertheless, we prove a dihedral rigidity theorem for a large class of hyperbolic polyhedra, where the leaf of the underlying warped product space is flat and the warping function is not strictly log-concave.

Dihedral rigidity for submanifolds of warped product manifolds

TL;DR

This paper proves Gromov's dihedral extremality and rigidity conjectures for a wide class of codimension-zero submanifolds with polyhedral boundary inside warped product spaces, including cases where the submanifold faces are neither orthogonal nor parallel to the radial direction. It develops a spinorial framework using a warped-product-derived potential in a Dirac operator and leverages index theory for polyhedral boundaries to obtain sharp curvature and dihedral-angle equalities under nonzero degree maps. The results cover radially convex domains in warped products and extend to hyperbolic manifolds (notably parabolic prisms in ) and certain direct-product spaces, yielding rigidity statements that force local isometries or product decompositions when curvature, dihedral, and boundary data saturate the comparisons. Overall, the work broadens the scope of dihedral rigidity from nonnegative-curvature leaves to hyperbolic settings and product geometries, with implications for scalar curvature, boundary geometry, and geometric-topological rigidity phenomena. The methods unify index-theoretic techniques with a spinorial approach to warped-product metrics and provide a robust framework for Stoker-type and related rigidity results in polyhedral geometry.

Abstract

In this paper, we prove a dihedral extremality and rigidity theorem for a large class of codimension zero submanifolds with polyhedral boundary in warped product manifolds. We remark that the spaces considered in this paper are not necessarily warped product manifolds themselves. In particular, the results of this paper are applicable to submanifolds (of warped product manifolds) with faces that are neither orthogonal nor parallel to the radial direction of the warped product metric. Generally speaking, the dihedral rigidity results require the leaf of the underlying warped space to have positive Ricci curvature and the warping function to be strictly log-concave. Nevertheless, we prove a dihedral rigidity theorem for a large class of hyperbolic polyhedra, where the leaf of the underlying warped product space is flat and the warping function is not strictly log-concave.
Paper Structure (7 sections, 12 theorems, 144 equations, 1 figure)

This paper contains 7 sections, 12 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dihedral rigidity for radially convex domains of warped product manifolds
  3. Manifolds with polyhedral boundary
  4. Comparisons on scalar curvature and mean curvature
  5. Dihedral rigidity of radially convex domains in warped product manifolds
  6. Dihedral rigidity of hyperbolic manifolds with polyhedral boundary
  7. Dihedral rigidity of direct product spaces

Key Result

Theorem 1.3

Suppose $M$ is an $n$-dimensional radially convex domain in a warped product manifold $(X\times I, g)$, where $g=dr^2+\varphi^2h$, such that If $(N, { \macc@depth1 \frozen@everymath{\mathgroup\macc@group} \macc@set@skewchar \macc@nested@a111{} } )$ is a Riemannian manifold with polyhedral boundary and $f\colon N\to M$ is a spin polytopeSee Definition def:polytopeMap below for the precis

Figures (1)

  • Figure 1: Examples of radially convex domains in $\mathbb{R}^2$ with $X=S^1$ and $\varphi(r)=r^2$

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 20 more