Differential geometry
Master 1 course, ENS de Lyon
January-April 2020

ANNOUNCEMENT: due to the COVID-19 (coronavirus) outbreak, the lectures and the exercise sessions are suspended as of March 16, 2020. Nevertheless, the course will continue at distance. The instructor and the teaching assistant will post the course material in the "Calendar" section below. The students are welcome to contact them via email at the following addresses in order to ask questions concerning the content of the course, and if needed schedule a video call:
  • Marco Mazzucchelli, marco.mazzucchelli@ens-lyon.fr
  • Chih-Kang Huang, chih-kang.huang@ens-lyon.fr


Schedule

Exams

  • Midterm: February 24, 2020, 1:30pm - 3:30pm, Amphi A
  • Final exam: April 21, 2020, 9:00pm - 12:00pm, at distance (instructions will be sent by email in due time)

Calendar (course, exercise sessions)

January 13: Riemannian metrics
Riemannian volume form
Divergence of a vector field
Riemannian gradient of a function
Laplace-Beltrami operator
Harmonic functions
Conformal diffeomorphisms
January 15: Existence of isothermal coordinates on Riemannian surfaces
Vector bundles
Connections
January 20: Linear connections
Existence and uniqueness of the Levi-Civita connections
January 22: Exercises:
January 27: Exercises:
Solutions:
January 29: Exercises:
February 3: Geodesics
Geodesic vector field and geodesic flow
Exponential map
Injectivity radius
Positivity of the injectivity radius of compact sets
February 5: Exercises:
February 10: Variational characterization of geodesics
Gauss Lemma
Short geodesics are the unique minimizers of the length function
February 12: Exercises:
February 17: Hopf-Rinow Theorem
Riemann tensor
February 19: Exercises:
March 9: Parallel transport
Flat Riemannian manifolds
Symmetries of the Riemann tensor
Ricci tensor
Scalar curvature
Einstein manifolds
Sectional curvature
March 19: Exercises:
March 16: Notes:
Riemannian submanifolds
Second fundamental form
March 18: Exercises:
Solutions:
March 20:
(15h30, Amphi A)
Notes:
Embedded surfaces in the Euclidean 3-space
Gaussian curvature
Gauss' Theorema Egregium
Einstein manifolds of higher dimension have constant scalar curvature
March 23: Notes:
Hopf Umlaufsatz
March 25: Exercises:
Solutions:
March 30: Notes:
Gauss-Bonnet's Theorem
April 1: Exercises:
Solutions:
April 6: Notes:
Hessian of the length functional
Conjugate points
Jacobi fields
April 8: Exercises:
Solutions:
April 15: Exercises:
Solutions:
UMPA ENS de Lyon Université de Lyon