Some references

The course does not  follow a particular textbook, we'll give some useful references on this page.

Bacground on analysis (function with bounded variation and absolutely continuous functions).

• Section 3.5 in the book Real analysis. Modern techniques and their applications. Second edition. by G. Folland.   A Wiley-Interscience Publication. John Wiley & Sons, Inc.
• Chapter 9, especially § 9.2, 9.3, 9.4 and 9.8 in Angus E. Taylor General theory of functions and integration. Blaisdell Publishing Co. Ginn and Co.  New York-Toronto-London 1965.
• Chapter 9, Sections 32-33 in the book by A. N. Kolmogorov   S. V. Fomin  Introductory Real Analysis (Dover Books on Mathematics).

Geometry of Metric Spaces

The following books are general introductions to metric geometry:

• A. Papadopoulos Metric spaces, convexity and nonpositive curvature,European Mathematical Society (EMS), Zürich, 2005.
• D. Burago, Y. Burago, S. Ivanov, A course in metric geometry.Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001.

On curves with finite total curvature

• Yu. G. Reshetnyak, The theory of curves in differential geometry from the point of view of the theory of functions of a real variable, Russian Math. Surveys 60 (2005), no. 6, 1165–1181.
• The book General theory of irregular curves. by Yu. G. Reshetnyak, A. D. Alexandrov,
  Kluwer Academic Publishers Group, Dordrecht, 1989.
• Yu. G. Reshetnyak, A. D. Alexandrov, Integral curvature of a curve in n-dimensional Euclidean space. Siberian Mathematical Journal, 1988, Volume 29-1, pp 1–16.
• John Sullivan, Curvatures of smooth and discrete surfaces. Chapter in the book Discrete differential geometry, Oberwolfach Semin., 38, Birkhäuser, Basel, 2008.

Hyperbolic Geometry

• F. Bonhaon, Low-dimensional geometry. From Euclidean surfaces to hyperbolic knots.
   American Mathematical Society, 2009.
• B. Martelli,  An Introduction to Geometric Topology. arXiv:1610.02592

Alexandrov Geometry

• Martin Bridson and André Haefliger Metric Spaces of Non-positive Cur-
  vature, Springer 1999.
• D. Burago, Yu. Burafo and S. Ivanov A course in Metric Geometry, AMS 2001 (chap. 9).
• S. Alexander, V. Kapovitch, A. Petrunin Invitation to Alexandrov geometry:
  CAT[0] spaces.  arXiv:1701.03483 
• S. Alexander, V. Kapovitch, A. Petrunin Alexandrov geometry: preliminary version arXiv:1903.08539. 
• S. Buyalo and V. Schroeder,  Spaces of curvature bounded above. Surveys in differential geometry. Vol. XI, 295–327.
• C. Plaut, Metric spaces of curvature ≥ k. Chap. 16 in Handbook of geometric topology, North-Holland, Amsterdam, 2002.

Minkowski, Funk and Hilbert Geometry

• Handbook of Hilbert Geometry, IRMA Lectures in Mathematics and Theoretical Physics, 22. European Mathematical Society, Zürich, 2014. Mainly chapters 1 and 2.
• A. Papadopoulos and M. Troyanov, Weak Finsler structures and the Funk weak metric.  
  Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 2, 419–437.
• H. Busemann and P. Kelly, Projective geometry and projective metrics. Academic Press Inc., New York, N. Y., 1953.

The first two references are downloadable from my webpage here.