Poincaré sphere

Riemannian Geometry — 1st Semester 2011/2012

Lecturer: José Natário
Email: jnatar@math.tecnico.ulisboa.pt
Office: Mathematics Building, 4th floor, room 4.13
Classes: Mondays from 11:00 to 13:00 and Thursdays from 9:00 to 11:00.
Office Hours: Wednesdays from 14:00 to 16:00.
About the picture: See this website and this article.


  • The grades for the make up test are available (check the side bar). If you want to see your test just drop by my office.
  • You can learn more about Lie groups and de Rham cohomology here (in Portuguese, by Rui Loja Fernandes).
  • You can find more information about the parallel postulate here (by Craig Kaplan).


  • Manifolds: differentiable manifolds; differentiable maps; tangent space; immersions and embeddings; vector fields; flows; Lie bracket; Lie groups; orientability; manifolds with boundary; differential forms; integration on manifolds; Stokes theorem; tensor fields.
  • Metrics: Riemannian manifolds; isometries; left-invariant metrics; affine connections; Levi-Civita connections; geodesics; minimizing properties of geodesics; Hopf-Rinow theorem.
  • Curvature: curvature tensor; sectional curvature; Ricci tensor; scalar curvature; connection and curvature forms; Cartan structure equations; index of a vector field at a singularity; Euler characteristic; Gauss-Bonnet theorem; isometric immersions; Gauss map; mean and Gauss curvatures; Theorema Egregium; first and second fundamental forms.
  • Applications: general relativity.


  • L. Godinho e J. Natário, An Introduction to Riemannian Geometry with Applications
  • Manfredo Perdigão de Carmo, Riemannian geometry, Birkhäuser, (1993)
  • Manfredo Perdigão de Carmo, Differential Forms and Applications, Springer (1994)
  • W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press (2003)

Grading Policy

  • Tests: There will be two tests each counting 35% towards the final grade (dates to be arranged). You will be able to make up for one of these tests the week after classes end.
  • Homework: There will be weekly problem sets making up 30% of the final grade. Late homework will not be accepted.


For more exercises check the course webpages from previous years (mostly in Portuguese):


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