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.

Annoucements

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)

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.

Week 1 (due on September 22)
Week 2 (due on September 29)
Week 3 (due on October 6)
Week 4 (due on October 13)
Week 5 (due on October 20)
Week 6 (due on October 27)
Week 7 (due on November 3)
Week 8 (due on November 10)
Week 9 (due on November 17)
Week 10 (due on November 25)
Week 11 (due on December 2)
Week 12 (due on December 12)
Week 13 (due on December 15)
Extra Homework 1 (optional, but must be handed in before the make up test for credit)
Extra Homework 2 (optional, but must be handed in before the make up test for credit)

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