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Local characterization of polyhedral spaces

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Abstract

We show that a compact length space is polyhedral if a small spherical neighborhood of any point is conic.

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Notes

  1. It also follows that \(V_i\) forms a strongly convex subset of \(X\); i.e., any minimizing geodesic in \(X\) with ends in \(V_i\) lies completely in \(V_i\). This property is not needed in our proof, but it is used in the alternative proof; see the last section.

References

  1. Akopyan, A.V.: A piecewise linear analogue of Nash–Kuiper theorem, a preliminary version (in Russian). www.moebiuscontest.ru

  2. Freedman, M.H.: The topology of four-dimensional manifolds. J. Differ. Geom. 17(3), 357–453 (1982)

    MATH  Google Scholar 

  3. Krat, S.: Approximation problems in length geometry. Thesis (2005)

  4. Lebedeva, N.: Number of extremal subsets in Alexandrov spaces and rigidity. Electron. Res. Announc. Math. Sci. 21, 120–125 (2014)

    MATH  MathSciNet  Google Scholar 

  5. Lebedeva, N.: Alexandrov spaces with maximal number of extremal points. Geom. Topol. (to appear) (2015). arXiv:1111.7253

  6. Milka, A.D.: Multidimensional spaces with polyhedral metric of nonnegative curvature. I. (Russian) Ukrain. Geometr. Sb. Vyp. 5–6, 103–114 (1968)

    MathSciNet  Google Scholar 

  7. Petrunin, A., Yashinski, A.: Piecewise distance preserving maps. St. Petersb. Math. J. (to appear). arXiv:1405.6606

  8. Zalgaller, V.A.: Isometric imbedding of polyhedra (Russian). Dokl. Akad. Nauk SSSR 123, 599–601 (1958)

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

We would like to thank Arseniy Akopyan, Vitali Kapovitch, Alexander Lytchak and Dmitri Panov for their help.

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Authors and Affiliations

  1. Steklov Institute, 27 Fontanka, St. Petersburg, 191023, Russia

    Nina Lebedeva

  2. Mathematics Department, St. Petersburg State University, Universitetsky pr., 28, Stary Peterhof, 198504, Russia

    Nina Lebedeva

  3. Mathematics Department, PSU, University Park, PA, 16802, USA

    Anton Petrunin

Authors
  1. Nina Lebedeva
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  2. Anton Petrunin
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Correspondence to Anton Petrunin.

Additional information

N. Lebedeva was partially supported by RFBR Grant 14-01-00062. A. Petrunin was partially supported by NSF Grant DMS 1309340.

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Lebedeva, N., Petrunin, A. Local characterization of polyhedral spaces. Geom Dedicata 179, 161–168 (2015). https://doi.org/10.1007/s10711-015-0072-x

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  • DOI: https://doi.org/10.1007/s10711-015-0072-x

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