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Metric geometry, fall 2013


Ilkka Holopainen


10 cu


Advanced studies



Metric geometry and analysis on metric spaces are nowadays very popular research topics. The purpose of this course is to introduce some basic notions of metric geometry (like various notions of curvature and boundaries of metric spaces), methods, and results.


Basic and intermediate studies.

Knowledge on Topology II, Real Analysis I, Introduction to Differential Geometry, and Riemannian Geometry would be useful but not necessary.


Weeks 36-49, Tuesday 12-14 in room B322, and Thursday 14-16 in room C123. Note: The last lecture is on Thursday, December 5.


The course can be passed by an exam or by solving home work problems and writing an essay.

Content (tentative)

  • Metric spaces, in particular length spaces
  • Model spaces
  • Alexandrov-spaces
  • Non-positively curved metric spaces
  • Metric spaces with curvature bounded from below
  • CAT(κ)-spaces
  • The Cartan-Hadamard theorem
  • Gromov-Hausdorff convergence
  • Gromov-hyperbolic metric spaces


  • M. Bridson, A. Haefliger: Metric Spaces of Non-Positive Curvature, Springer, 1999.
  • D. Burago, Y. Burago, S. Ivanov: A Course in Metric Geometry, American Mathematical Society, 2001.
  • Gromov: Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser, 1999.


Lecture notes: I. Holopainen: Metric Geometry.


Did you forget to register?  What to do?


1.Wednesday 14-16 B322 Juhani Koivisto

Home work assignments and solutions 

Exercise 1  Solutions 1

Exercise 2  Solutions 2

Exercise 3  Solutions 3

Exercise 4  Solutions 4

Exercise 5  Solutions 5

Exercise 6  Solutions 6

Exercise 7  Solutions 7

Exercise 8  Solutions 8

Exercise 9  Solutions 9

Exercises 10  Solutions 10

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