Metric geometry

Textbooks

• Burago, Burago, Ivanov, 2001: A course in metric geometry (pdf, errata )
  • An accessible introduction and Steele Prize winner
  • However, contains many errors, some serious—check the errata!
• Bridson & Haefliger, 1999: Metric spaces of non-positive curvature (doi)
  • Part I covers the basic concepts of metric geometry: length spaces, geodesic spaces, the “model spaces” of constant curvature, etc.
  • Surprisingly readable and much more carefully written than Burago et al
  • On the whole, my preferred introduction to metric geometry
• Papadopoulos, 2014: Metric spaces, convexity and nonpositive curvature, 2nd ed. (doi)
  • A careful, systematic book with detailed proofs
  • Chapter 3 covers maps between metric spaces, especially nonexpansive maps, drawing on work by Busemann, such as: Busemann, 1964: Length-preserving maps (pdf)

Other books, monographs, and lecture notes

• Gromov, 1991: Metric structures for Riemannian and non-Riemannian spaces (doi)
  • Influential book by Mikhail Gromov, translated from 1981 French original
  • Burago et al: “a remarkable book, which gives a vast panorama of ‘geometrical mathematics from a metric viewpoint.’ Unfortunately, Gromov’s book seems hardly accessible to graduate students and non-experts in geometry.”
• Ballmann, 1995: Lectures on spaces of nonpositive curvature (doi, pdf)
  • Ballmann, Gromov, Schroeder, 1985: Manifolds of nonpositive curvature (doi)
• Shioya, 2016: Metric measure geometry: Gromov’s theory of convergence and concentration of metrics and measures (arxiv, doi, pdf)
  • Self-described as an expansion on Chapter \(3 \tfrac{1}{2}\) of Gromov’s book
• Alexander, Kapovitch, Petrunin, 2019+, draft: Alexandrov geometry (arxiv)

Metric measure spaces

• Villani, 2009: Optimal transport: Old and new, Chapter 27: Convergence of metric-measure spaces (doi)
• Sturm, 2006, in Acta Math.: On the geometry of metric measure spaces, I (doi) and II (doi)
• Sturm, 2012: The space of spaces: Curvature bounds and gradient flows on the space of metric measure spaces (arxiv)
  • Talk: Geometric analysis on the space of metric measure spaces (video)

Metric functionals, aka horofunctions or Busemann functions

• Bridson & Haefliger, 1999, Ch. II.8: The boundary at infinity of a CAT(0) space
• Karlsson, 2019: Elements of a metric spectral theory (arxiv, pdf)
  • Proposes metric analogs of linear functionals, eigenvalues, and other elements of linear spectral theory
• Karlsson, 2020: Hahn-Banach for metric functionals and horofunctions (arxiv)

Metric graphs, aka metrized graphs

Metric geometry makes contact with graph theory through metric graphs.

• Burago et al, 2001, Sec. 3.2.2: Metric graphs
• Baker & Faber, 2006: Metrized graphs, Laplacian operators, and electrical networks (pdf, arxiv)
  • Expository paper in a book on quantum graphs (doi)
  • Laplacians on metric graphs interpolate between classical Laplacians, which are differential operators, and graph Laplacians
• Baker & Rumely, 2007: Harmonic analysis on metrized graphs (doi, arxiv)

Metric embeddings

Questions about embedding metric spaces into Euclidean spaces and other normed spaces are a staple of discrete and combinatorial geometry, with applications such as kernels in machine learning and the Johnson-Lindenstrauss lemma.

• Deza & Laurent, 1997: Geometry of cuts and metrics (doi, pdf)
  • Sec 6.2: “Characterization of \(L_2\)-embeddability” bears on the problem of constructing kernels from metrics, as explained in a blog post by Suresh Venkatasubramanian
• Matoušek, 2002: Lectures on discrete geometry, Chapter 15: Embedding finite metric spaces into normed spaces
• Matoušek, 2013: Lecture notes on metric embeddings (pdf)
  • Frequently referenced in Moses Charikar’s 2018 lecture notes for Stanford’s CS 369M: “Metric embeddings and algorithmic applications”

See also Harald Räcke’s 2006 lecture notes on metric embeddings.

Miscellany

• Tuzhilin, 2016: Who invented the Gromov-Hausdorff distance? (arxiv)
  • Argues that mathematician David Edwards deserves some of the credit for the Gromov-Hausdorff distance
  • Edwards, 1975: The Structure of Superspace (doi, pdf)
• Existence of measures invariant under isometries (Math.SE )