Tamás Darvas
Assistant Professor, University of Maryland

Class: TTh 2:00 PM - 3:15 PM, MTH B0431.

Office hours: TTh 1:00 PM - 2:00 PM (or by appointment), Math 4416.

Homeworks: assgined weekly, grades maintained in ELMS.

• Homework 1: Prove Lemma 1.13. Solve the following exercises at the end of Chapter 1: 1-2,1-3,1-4,1-6,1-9,1-11,1-12.


• Homework 2: Solve the following exercises at the end of Chapter 2: 2-4,2-5,2-6,2-10,2-14, Chapter 3: 3-3,3-4,3-6,3-8.


• Homework 3: Solve the following exercises at the end of Chapter 4: 4-4,4-5,4-6,4-10, Chapter 5: 5-1,5-6,5-11,5-17,5-18,5-20.


• Homework 4: Solve the following exercises at the end of Chapter 8: 8-3, 8-13, 8-14, 8-16, 8-19 Chapter 9: 9-5, 9-7, 9-8, 9-16, 9-18.


• Homework 5: Solve the following exercises at the end of Chapter 10: 10-1(a,b),10-5, 10-6, Chapter 11: 11-1, 11-5, 11-6, 11-7, Chapter 14: 14-1, 14-5.


• Homework 6: Solve the following exercises at the end of Chapter 2: 2-9, 2-12, 2-13, 2-14, 2-21, 2-22, 2-23 (solve the last two problems only for orientable Riemannian manifolds. Hint: use Stokes theorem).


• Homework 7: Solve the following exercises at the end of Chapter 3: 3-2 (follow the book's definition of frame-homogeneity, that is slightly different compared to the lectures), 3-3, 3-7, 3-19, Chapter 4: 4-2, 4-8, 4-10.


• Homework 8: Solve the following exercises at the end of Chapter 5: 5-3, 5-4, 5-7(a), 5-10, 5-11,5-13 (ignore parts of problems related to pseudo-Riemannian metrics).


• Homework 9: Solve the following exercises at the end of Chapter 6: 6-2, 6-4, 6-10, 6-23, Chapter 7: 7-2,7-4.



Lecture log, about what was covered from Lee's textbooks:
Introduction to Smooth manifolds:

• Chapter 1 (read last section on manifolds with boundary as part of your homework).


• Chapter 2.


• Chapter 3.


• Chapter 4 (Skip Section "Smooth Covering maps").


• Chapter 5.


• Chapter 6 (only read statements of Sard's theorem, Whitney's embedding/immersion theorems).


• Chapter 8 (skip the Section "The Lie Algebra of a Lie Group" and thereafter).


• Chapter 9 (skip Sections "Flowouts" and "Flows and Flowouts ...", also skip Section "Time-Dependent vectorfields" and thereafter).


• Chapter 10 (skip everything after definition of bundle homomorphism).


• Chapter 11 (skip the Section "Restricting Covector Fields to Submanfolds" and thereafter).


• Chapter 12 (skip Section "Lie Derivatives of Tensor Fields" and thereafter).


• Chapter 14 (skip Section "Exterior Derivatives and Vector Calculus in R^3" and thereafter).


• Chapter 15 (skip Section "The Riemannian volume form" and thereafter).



Introduction to Riemannian manifolds:

• Chapter 2 (skip Riemannian submersions, Riemannian coversings, Pseudo-Riemannian manifolds).


• Chapter 3 (skip Invariant metrics on Lie groups and thereafter).


• Chapter 4 (skip Pullback connections).


• Chapter 5 (skip Tubular neighborhoods and Fermi coordinates).


• Chapter 6 (skip material after corollaries to the Hopf-Rinow theorem).


• Chapter 7 (skip Weyl Tensor and thereafter).


• Chapter 8 (skip Computations in Semigeodesic Coordinates, Minimal Hypersurfaces).



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Previous semesters: Fall 2018 MATH 868C, Fall 2017 MATH 131, Spring 2017 MATH 430, Fall 2016 MATH 868D, Fall 2015 MATH 220, MATH 401, Spring 2015 MATH 461, Fall 2014 MATH 430.