MATH 621: Differential Geometry (Spring 2025)
Tu Th 8:30 AM - 9:45 AM Physics 227
Instructor: Saman Habibi Esfahani Email: Saman.HabibiEsfahani@duke.edu
Office Hours: Tuesdays & Thursdays 10:30am - 11:30am, office: 106, Physics Building
Midterm: Feb 27, 8:30 am Final exam: April 15, 8:30am
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Lecture notes (Final version, updated APRIL 11)
Recorded Videos: Playlist
Session 1, (Smooth manifolds)
Session 2, (Derivative and tangent spaces)
Session 3 (part I), (Submanifolds, immersions, and embeddings)
Session 3 (part II), (Equivalent definitions of submanifolds)
Session 4, (Whitney embedding theorem)
Session 5, (Tensors and multi-linear maps)
Session 6, (Differential forms)
Session 7, (Riemannian manifolds)
Session 8, (Existence of Riemmanian metrics)
Session 9, (Covariant derivative)
Session 10, (The Levi-Civita connection)
Session 11, (The Geodesics)
Session 12, (The Riemann curvature tensor)
Session 13, (Minimal submanifolds)
Session 14, (The holonomy group)
Session 15, (Berger's classification theorem)
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Homework: There will be weekly homework due on Fridays (11:59 pm). I will post the homework here (and on Gradescope) each Friday, and you should upload your homework on the Gradescope page of the course. Typing in Latex is recommended.
Homework 1 (due date Jan 31) -- Solutions can be found in the textbook.
Homework 2 (due date Feb 07) -- Solutions can be found in the textbook.
Homework 3 (due date Feb 14) -- Solutions can be found in the textbook.
Homework 4 (due date Feb 21) -- Solutions can be found in the textbook.
Homework 5 (due date Feb 28) -- Solutions can be found in the mentioned reference.
Midterm
Homework 6 (due date Mar 21) -- Solutions can be found in the mentioned reference (Walpuski).
Homework 7 (due date Mar 28) -- Solutions can be found in the mentioned reference (Colding-Minicozzi).
Homework 8 (due date Apr 04) -- Solutions can be found in the mentioned reference (Audin-Damain).
Homework 9 (due date Apr 18) (last homework) -- Solutions can be found in the mentioned reference (Joyce).
Syllabus:
Smooth manifolds, tangent and cotangent spaces, submanifolds, differentiable maps, immersions, submersions, embeddings, tensor calculus, tensors, differential forms, de Rham cohomology, Riemannian metrics, geodesics, vector and principal bundles, connections, Levi-Civita connection, curvature tensors, Riemann curvature tensor, Ricci tensor, sectional curvature, minimal surfaces, symplectic geometry, Kähler geometry, complex manifolds, Hodge theory, characteristic classes, harmonic maps, heat flow methods, eigenvalues of the Laplacian, gauge theory, Yang-Mills theory, Morse theory, Floer theory, manifolds with special holonomy, calibrated submanifolds.
References:
Main textbook (which we use for the foundations):
Riemannian Geometry and Geometric Analysis, 7th edition, Jürgen Jost
Some other references we might use:
Connections, Curvature, and Yang-Mills theory: Differential Geometry III: Gauge Theory by Thomas Walpuski
Special holonomy manifolds: Compact Manifolds with Special Holonomy by Dominic Joyce
Morse Theory by John Milnor.
A Course in Minimal Surfaces by Tobias Colding and William Minicozzi
Exams: There will be in class midterm and final.
Grading: Class activity (25%) + Homework (25%) + Midterm(25%) + Final (25%)