 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; leftinvariant metrics; affine connections; LeviCivita connections; geodesics; minimizing properties of geodesics; HopfRinow 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; GaussBonnet 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.
For more exercises check the course webpages from previous years (mostly in Portuguese):
