%From branbeard@hotmail.com  Fri Apr  8 20:36:29 2005
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\begin{center}{Oral Qualifying Exam Syllabus \\
Brian Manning}
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MAJOR TOPIC: RIEMANNIAN GEOMETRY

Preparation: Manifolds, vector bundles, tensors, differential
forms

Riemannian metrics

Connections, covariant derivative, parallel transport

Exponential map, geodesics, Gauss Lemma

Convex neighborhoods

Completeness, geodesic completeness; Hopf-Rinow Theorem

Sectional Curvature, Ricci Curvature, and Scalar Curvature

First and second variation formulas

Jacobi Fields

Submanifolds and immersions

Distance functions, curvature equations

Classification of constant curvature spaces

First and second comparison estimates

Hadamard-Cartan theorem, Bonnet-Myers theorem

Conjugate points, conjugate radius, injectivity radius

Fundamental group and nonpositive curvature

Hypersurfaces in Riemannian manifolds

Synge's theorem

Lie Groups, bi-invariant metrics

Free isometric group action

Riemann submersion formula, homogeneous spaces

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References: Karsten Grove, Riemannian Geometry: A Metric Entrance

        Peter Petersen, Riemannian Geometry

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MINOR TOPIC: ALGEBRAIC TOPOLOGY

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Covering spaces, path lifting

Homotopy, homotopy lifting theorem

Fundamental group,
Van Kampen Theorem

Singular homology group, chain complexes,
Homotopy invariance

Relative homology

Exact sequences,
Excision theorem,
Mayer-Vietoris sequences

Betti numbers and Euler characteristics

Orientation of a manifold

Singular cohomology,
Cup and cap products

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References: Allen Hatcher, Algebraic Topology \\
        James Vick, Homology Theory: An Introduction to Algebraic Topology

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