SYLLABUS FOR RICARDO MENDES' ORAL EXAM

MAJOR AREA: DIFFERENTIAL GEOMETRY
-Riemannian metrics
-Levi-Civita connection
-Curvature tensor, sectional, scalar, Ricci curvatures
-Geodesics, exponential map, Gauss Lemma
-Surfaces of revolution, Clairaut's Relation
-Jacobi Fields, conjugate points, focal points
-Hopf-Rinow Theorem
-Spaces of constant curvature
-Isometric Immersions, second fundamental form
-Riemmanian Submersions, O'Neill tensor, complex projective space
-Formulas for first and second variations of energy
-Bonnet-Myers, Synge-Weinstein and Hadamard's Theorems
-Rauch Comparison Theorems
-Toponogov Theorem
-Holonomy, DeRham Decomposition Theorem
-Homogeneous and Symmetric Spaces
-Bishop-Gromov Volume Comparison

MINOR AREA: ALGEBRAIC TOPOLOGY
-Fundamental Group, Van Kampen's Theorem
-Covering Spaces: Deck Transformations, classification
-Homology and cohomology: Singular, cellular, simplicial homologies;
Mayer-Vietoris, Universal Coefficient Theorems, Kunneth Formula, cup
and cap products, Poincare Duality
-Brouwer and Lefschetz Fixed Point Theorems
-CW complexes: cellular approximation, CW aproximation
-Fibrations, Cofibrations and how to replace a map by them (Mapping
Cylinder and Path Space)
-Higher Homotopy Groups: Long Exact Sequence of Fibration, Hurewicz
and Whitehead Theorems, Homotopy Excision, Freudental Suspension
Theorem.
-Eilenberg-MacLane Spaces, Postnikov Towers
-Serre Spectral Sequence