Riemannian Geometry I
Monday 9:00-12:10
Lectures will be online on zoom starting on Monday 5.9. at 9 am. Lectures will be recorded.
Zápočet: there will be 9 homeworks during semester, each week will be given at most one homework. As a requirement to take the final exam students must accomplish 20% of all possible points from homeworks (each homework has to be submitted in two weeks if not specified otherwise).
Exam: if a student accomplishes 80% of all possible points from homeworks, then he or she gets 1 from the final exam. Otherwise the final exam will be oral. Student can decide whether he/she wants to solve problems related to the course or be evaluated from theory. Before taking exam each student needs to obtain zápočet.
Requirements for oral exams:
I can ask any theorem or statement we covered on lectures. I can ask the following definitions and proofs.
Definitions:
topological and smooth manifold, tangent space, tangent vector, tangent bundle, vector field, Lie bracket and its properties, flow of vector field and integral curves, complete vector fields, tensors and operations on them, tensor fields and operations on vector fields, Lie derivative of tensor field, Riemannian metric, distance on Riemannian manifold, affine connection, Christoffel symbols, torsion of affine connection, parallel transport, geodesic equation, Levi-Civita connection, functionals of length and energy, curvature of affine connection, Riemannian curvature and its properties, Gauss curvature, Ricci and scalar cuvature, sectional curvature, exponential map, normal coordinates
Proofs:
The existence and uniqueness of Levi-Civita connection, symmetries of Riemannian curvature, determination of Riemannian curvature by sectional curvature
Score from HW
Lecture notes:
2. week video slides Homework1 due to 26.10
3. week video slides Homework2 due to 2.11.
4. week video slides Homework3 due to 9.11.
5. week video slides Homework4 due to 16.11.
6. week video slides Homework5 due to 23.11.
7. week video slides Homework6 due to 30.11.
8. week video slides No homework this week
9. week video slides Homework7 due to 14.12.
10. week video slides Homework8 due to 21.12.
11. week video slides Homework9 due to 4.1.
Recommended literature
1) O. Kowalski, Základy Riemannovy geometrie, skripta, 2. vydání, vydavatelství Karolinum, 2001. (in Czech)
2) P. do Carmo, Riemannian Geometry 1, Birkhaeuser.
3) M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1 - 2., Publish or Perish Inc..
4) P. Petersen, Riemannian Geometry, Springer, Graduate Texts in Mathematics, Vol. 171, 2nd Edition, 2006.
5) W. Curtis, F. Miller, Differential Manifolds & Theoretical Physics, Pure and Applied Math.
6) S. Kobayashi and K. Nomizu, Foundations of Differential geometry I, II, Interscience Publishers 1963, 1969.
7) S. Helgason, Differencial´naja geometrija i simmetričeskije prostranstva (překlad z angličtiny), Izd. MIR, Moskva 1964 (Kapitola 1).
8) S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic press, 1978.
9) R. L. Bishop, R. J. Crittenden, Geometry of Manifolds, AMS Chelsea Publishing, 2001.
10) L. Nicolaescu, Lectures on the Geometry of Manifolds, World Scientific Publishing Company, 2007. (Available at https://www3.nd.edu/~lnicolae/Lectures.pdf)
11) I. Kolář, P. W. Michor, J. Slovák, Natural operations in differential geometry,Springer, 1993.
12) R.W. Sharpe, Differential geometry - Cartan’s generalization of Klein’s Erlangen program, Springer, 1997.