### 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.