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On Riemannian polyhedra with non-obtuse dihedral angles in 3-manifolds with positive scalar curvature

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We determine the combinatorial types of all the 3-dimensional simple convex polytopes in \({\mathbb {R}}^3\) that can be realized as mean curvature convex (or totally geodesic) Riemannian polyhedra with non-obtuse dihedral angles in Riemannian 3-manifolds with positive scalar curvature. This result can be considered as an analogue of Andreev’s theorem on 3-dimensional hyperbolic polyhedra with non-obtuse dihedral angles. In addition, we construct many examples of such kind of simple convex polytopes in higher dimensions.

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Acknowledgements

The author wants to thank Jiaqiang Mei, Yalong Shi, Xuezhang Chen and Yiyan Xu for some valuable discussions on Riemannian geometry.

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  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Li Yu

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  1. Li Yu
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This work is partially supported by National Natural Science Foundation of China (Grant No. 11871266) and the PAPD (priority academic program development) of Jiangsu higher education institutions.

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Yu, L. On Riemannian polyhedra with non-obtuse dihedral angles in 3-manifolds with positive scalar curvature. manuscripta math. 174, 269–286 (2024). https://doi.org/10.1007/s00229-023-01501-7

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