Saddle Codazzi tensor on spheres and hemisphere rigidity
Abstract
Let (Mn,g), n≥ 3, be a compact connected Riemannian manifold with smooth boundary. We ask whether M must be isometric to the standard hemisphere if secg≥ 1, $\partial M $ is isometric to the standard unit sphere, and its second fundamental form is Codazzi. Our approach centers on saddle Codazzi tensors on round spheres, where the saddle condition requires the restriction to every tangent two-plane to have nonpositive determinant. By analyzing the associated support maps, we prove that every smooth saddle Codazzi tensor on 𝕊m vanishes identically for m≥ 3. Together with Panina's examples on 𝕊2, this shows that, among unit spheres of dimension at least two, nontrivial smooth saddle Codazzi tensors exist precisely in dimension two. The vanishing theorem yields an affirmative answer to the hemisphere rigidity question for n≥ 4. In dimension three, we realize the nontrivial saddle Codazzi tensors provided by Panina's construction, after scaling, as the second fundamental forms of smooth perturbations of the standard hemisphere, preserving the round boundary metric and the sectional curvature lower bound. This produces counterexamples to rigidity in dimension three. Nevertheless, under the same curvature and boundary assumptions, we prove the sharp inradius estimate rin(M)≤ π/2, with equality if and only if M is isometric to the standard hemisphere.
Related
- Strict-Saddle Landscapes and Multi-Rank Geometry in Low-Tubal-Rank Tensor Sensing
- Length spectrum rigidity and flexibility of spheres of revolution with one equator
- Rigidity of Minimal Submanifolds in Spheres of Higher Codimension
- Rigidity of unit spheres in finite-dimensional real normed spaces
- A geometric rigidity estimate for codimension-1 immersions into spheres
BibTeX
@misc{ge2026saddle,
title = {{Saddle Codazzi tensor on spheres and hemisphere rigidity}},
author = {Jian Ge and Mijia Lai and Yuntian Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25035},
eprint = {2610.25035},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25035},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Saddle Codazzi tensor on spheres and hemisphere rigidity
AU - Jian Ge
AU - Mijia Lai
AU - Yuntian Zhu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25035
UR - https://arxiv.org/abs/2610.25035
AB - Let (M^n,g), n≥ 3, be a compact connected Riemannian manifold with smooth boundary. We ask whether M must be isometric to the standard hemisphere if sec_g≥ 1, $\partial M $ is isometric to the standard unit sphere, and its second fundamental form is Codazzi. Our approach centers on saddle Codazzi tensors on round spheres, where the saddle condition requires the restriction to every tangent two-plane to have nonpositive determinant. By analyzing the associated support maps, we prove that every smooth saddle Codazzi tensor on 𝕊^m vanishes identically for m≥ 3. Together with Panina's examples on 𝕊^2, this shows that, among unit spheres of dimension at least two, nontrivial smooth saddle Codazzi tensors exist precisely in dimension two. The vanishing theorem yields an affirmative answer to the hemisphere rigidity question for n≥ 4. In dimension three, we realize the nontrivial saddle Codazzi tensors provided by Panina's construction, after scaling, as the second fundamental forms of smooth perturbations of the standard hemisphere, preserving the round boundary metric and the sectional curvature lower bound. This produces counterexamples to rigidity in dimension three. Nevertheless, under the same curvature and boundary assumptions, we prove the sharp inradius estimate r_in(M)≤ π/2, with equality if and only if M is isometric to the standard hemisphere.
ER -