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.