# Differential Analysis & Topology

### Topology and/or Differential Geometry Topic List

For the Oral Qualifying Exam

For the oral qualifying exam in Topology and/or Differential Geometry the candidate is to prepare a syllabus by selecting topics from the list below. The total amount of material on the syllabus should be roughly equal to that covered in a standard one-semester graduate course. Once you have made your selections discuss them with the professor who will examine you.

#### Topology

• Basic topological notions: path connectivity, connectivity, product topology, quotient topology.
• The fundamental group, computation of the fundamental group, van Kampen's theorem, covering spaces.
• Homology: singular chains, chain complexes, homotopy invariance, relationship between the first homology and the fundamental group, relative homology, the long exact sequence of relative homology, the Mayer-Vietoris sequence, applications to computing the homology of surfaces, projective spaces, etc.
• Topological manifolds, differentiable manifolds.

#### Differential Geometry of Curves and Surfaces in Euclidean Space

• The orthogonal group in 2 and 3 dimensions, the Serret-Frenet frame of a space curve.
• The Gauss map and the Weingarten equation for a surface in Euclidean 3-space, the Gauss curvature equation and the Codazzi-Mainardi equation for a surface in Euclidean 3-space.
• The surfaces in Euclidean 3-space of zero Gauss curvature.
• The fundamental existence and rigidity theorem for surfaces in Euclidean space.
• The Gauss-Bonnet formula for surfaces in Euclidean 3-space.

#### Differential Geometry of Riemannian Manifolds

• Riemannian metrics and connections.
• Geodesics and the first and second variational formulas.
• Completeness and the Hopf-Rinow theorem.
• The Riemann curvature tensor, sectional curvature, Ricci curvature, and scalar curvature.
• The theorems of Hadamard and Bonnet-Myers.
• The Jacobi equation.
• The geometry of submanifolds – the second fundamental form, equations of Gauss, Ricci, and Codazzi.
• Spaces of constant curvature.

### References

• Harper and Greenberg, Algebraic Topology, a First Course, Parts I and II
• M. do Carmo, Differential Geometry of Curves and Surfaces
• M. do Carmo, Riemannian Geometry
• M. Spivak, A Comprehensive Introduction to Differential Geometry
• S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces
• S. Sternberg, Lectures on Differential Geometry, 2nd ed.