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Alexandrov geometry: foundations

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arxiv 1903.08539 v6 pith:BHA54JIQ submitted 2019-03-20 math.DG math.MG

Stephanie Alexander , Vitali Kapovitch , Anton Petrunin This is my paper · ORCID
classification math.DGmath.MG
keywords alexandrovspacesboundedcurvatureaboveapplicationsaxiomsbelow
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Alexandrov spaces are defined via axioms similar to those given by Euclid. The Alexandrov axioms replace certain equalities with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above and curvature bounded below. The definitions of the two classes of spaces are similar, but their properties and known applications are quite different. Our approach is novel in its attention to the interrelatedness of the two fields, and its emphasis on the way each illuminates the other. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering the two contexts simultaneously. We only consider the intrinsic theory, leaving applications aside. This book includes material up to the definition of dimension. Another volume still in preparation will cover further topics.

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Cited by 2 Pith papers

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    math.AP 2026-07 accept novelty 5.0 of 10

    Energy-minimizing harmonic maps from RCD(K,N) domains into small balls in CAT(κ) spaces are locally Lipschitz, completing the singular Bochner–Eells–Sampson picture.

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