For doctoral students; Semester 2, Thu. Period 3 · Graduate School of Science and Technology, Niigata University
Note
This course is held online via Zoom.
Textbooks
John W. Milnor, “Morse Theory”, Princeton University Press
Course contents
- Covariant differentiation, part 1
- Covariant differentiation, part 2
- The curvature tensor
- Geodesics and completeness, part 1
- Geodesics and completeness, part 2
- Geodesics and completeness, part 3
- The path space of a smooth manifold
- The energy of a path
- The Hessian of the energy function at critical paths
- Jacobi fields: the null space of $\mathrm{Hess}_{\gamma}(E)$, part 1
- Jacobi fields: the null space of $\mathrm{Hess}_{\gamma}(E)$, part 2; the index theorem, part 1
- The index theorem, part 2
- A finite-dimensional approximation to $\Omega^c$, part 1
- A finite-dimensional approximation to $\Omega^c$, part 2; the topology of the full path space, part 1
- The topology of the full path space, part 2
- Existence of non-conjugate points
- Some relations between topology and curvature, part 1
- Some relations between topology and curvature, part 2