Cyclical Deformations of Quadrangles in S 2κ and Defining Properties of Alexandrov Spaces

Abstract

This paper is mainly devoted to proving the four equivalent defining properties of a CAT(κ) space. The proof is based on an interesting tool we established which describes the cyclical five-step deformation procedure for quadrangles in the modal 2-plane S 2κ , including the limit shape of each step. As a byproduct we give a complete list of cyclical deformation procedures for all kinds of quadrangles on S 21 . At last we make a contrast of geometric properties of CAT with CBB spaces, including a comparison between their defining properties and a discussion about Alexandrov’s Lemma.

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Acknowledgements

The author would like to thank the hospitality of Hill Center, Mathematics Department of Rutgers University during his visiting there.

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Correspondence to Qing Song Cai.

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Supported by China Scholarship Council (Grant No. 201708120014)

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Cai, Q.S. Cyclical Deformations of Quadrangles in S 2κ and Defining Properties of Alexandrov Spaces. Acta. Math. Sin.-English Ser. (2021). https://doi.org/10.1007/s10114-021-9462-1

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Keywords

  • Spherical quadrangle
  • Alexandrov spaces
  • CAT spaces
  • CBB spaces

MR(2010) Subject Classification

  • 51M09
  • 53C23