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