Topics

Differentiable Manifolds, Smoothings and Triangulations
Diffeomorphism, Embeddings, Immersions Submersions
Tangent Vectors and Tangent Spaces
Transversality and Singularities of Maps
Degree Theory
Vector Fields and Index Theory
Vector Bundles and Fibre Bundles
Differentiable Forms
Rudiments of De Rham Theory
Riemannian Metrics
Riemannian Connections
The Curvature Tensor and its Contractions
Properties of Ricci and Scalar Curvature
Geodetics
Relations between Curvature and Homology

To get sample of problems you should be able to solve after having taken the course jump here!  

Bibliography

Hirsch, Differential Topology, Springer.
Bott, Tu Differential Forms in Algebraic Topology, Springer
Berger, Differential Geometry, Springer


Exams

There will be five exams divided into three sessions:  2 in Summer (June-July) , 2 in Autumn (September-October) and 1 in Winter (February).
Students can take at most one exam per session.
The exam will consist of a written exam and, possibly, an oral exam.
The maximum score in the written exam will be 24.  Once the written exam is passed students may choose whether to keep
the score they obtained or to afford the oral exam, in this case the score will be based on the performance in the oral exam only.
(e.g. If you get 24 in the written exam, you decide to take the oral one and  you perform poorly  you will be failed).

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