Lecturer

Tom Ilmanen

Coordinator

Valentin Bosshard

Lectures

Mo 14-16 CAB G 11
Mi 14-16 HG G 5

Forum

Link

Content

Submanifolds of \(\mathbb R^n\), immersions, submersions, and embeddings, Sard's Theorem, abstract differentiable manifolds, charts, vector fields and flows, vector bundles, tensor fields, covariant derivatives, parallel transport, Riemannian metrics, geodesics, Riemann curvature tensor. Complete manifolds, Hopf-Rinow theorem. Many examples including curves, surfaces, hyperbolic space, \(S^3\), the unit quaternions, the Gauss-Bonnet theorem, etc.

Lectures

Week	Date	Topics
1	20.09	Introduction and motivation for differential and Riemannian geometry. Regular curves in \(\mathbb R^n\), their arclength, unit tangent vector and curvature vector. Specialize to curves in \(\mathbb R^2\).
2	25.09.	More on curves in \(\mathbb R^2\): curvature of a curve determines curve up to rigid motion, rotation number. Curves in \(\mathbb R^3\): binormal vector, Frenet frame, torsion vector.
	27.09.	Statement of Fenchel's and Milnor's Theorem. Surfaces in \(\mathbb R^3\): Overview how to define curvature of surfaces. (A) Geometric definition of the curvature using curvature of curves.
3	02.10.	End of proof that definition (A) is well-defined. (B) Definition of curvature using Hessian of a function in well-chosen coordinates, 2nd fundamental form. Examples.
	04.10.	Principal directions for surface with symmetry. Vector fields on a surface. (C) Definition of curvature using directional derivative of the normal, the Weingarten map.
4	09.10.	Proof of equivalence of formulas from (B) and (C). Eyeglasses.
	11.10.	Overview of (D), (E): Formulas for parametrized surfaces and graphs. Intrinsic geometry, first fundamental form, (local) isometries. 3 Theorems: Theorema Egregium, Gauss-Bonnet Theorem, Uniformization Theorem.
5	16.10.	Overview of differentiable manifolds. Topological spaces. Charts, parametrizations, overlaps and transition maps, compatible charts.
	18.10.	Atlas, lots of examples including different atlases on spheres, the real projective space. Maximal atlas and theorem of existence. Definition of a differentiable manifold.
6	23.10.	Proof of maximal atlas theorem. Cunstruction of smooth manifolds: open subsets, product manifolds. Smooth maps, diffeomorphisms. Motivation abtract tangent vectors.
	25.10.	Four characterizations of tangent vectors. Definition of the tangent space at a point.
7	30.10.	Differential of a map, coordinate expression for differentials. Chain rule. The tangent bundle of a manifold.
	01.11.	Vector fields, number of pointwise linearly independent vector fields. Submanifolds. Next goal: When is the image or the preimage of a smooth map a submanifold?
8	06.11.	Immersions, submersions, embeddings, local diffeomorphisms. Covering maps. Properly discontinuous and free actions produce covering maps.
	08.11.	Orientations, orientation double cover.
9	13.11.	Local immersion theorem. Embedding criterion. Proper maps.
	15.11.	Sketch of steps to prove embedding theorem for proper maps by stating relevant lemmas. Countability definitions: second countable, paracompact, sigma-compact and relations between them. Ordinals.
10	20.11.	Bump functions, cutoff functions, partitions of unity. Different flavours of the Whitney embedding theorem.
	22.11.	Proof of the Whitney embedding theorem for compact manifolds. The submersion theorems. Critical point, critical values.
11	27.11.	Sard's Theorem. Derivatives of vector fields, Lie bracket.
	29.11.	Properties of the Lie bracket. Lie algebras. Flows: Definition, existence and uniquness theorems.
12	04.12.	Complete vector fields, maximum interval of existence, sublinear vector fields are complete. Group property of flows.
	06.12.	No class.
13	11.12.	Incomplete flows: Group property, domain of definition is open. Straightening vector fields. Definition pullback and pushforward of vector fields.
	13.12.	Properties of pullback and pushforward. Definition Lie derivative.
14	18.12.	Lie derivative equals Lie bracket.
	20.12.	Commuting flows and vector fields.Commutation error. Parking

Script:

• Old INOFFICIAL script from Ilmanen's lecture 2014. This is NOT the script of this lecture. The content of the lecture will add/skip/change many sections.
• Some students write collaboratively a script for this semester. Link to current version of the script, Link to report typos. If you want to join the project, get in touch with the coordinater of the lecture, Valentin.

Exercises

The new exercise sheets will be uploaded on Wednesdays. You have time until the following Wednesday at 18.00 to upload your solutions using the SAM upload tool (Can only be done when connected to the ETH-network).

The first exercise classes will take place in the first week, on Th 21.09 or Fr 22.09.

Exercise classes

Literature

Differential Geometry:

• John M. Lee: Introduction to Smooth Manifolds (2002).
• John M. Lee: Introduction to Riemannian Manifolds (2018).

Topology:

• Allen Hatcher: Notes on Introductory Point-Set Topology.
• Klaus Jänich: Topologie (2005) Chapter 1 (intuitive introduction to topological spaces and manifolds. In German).
• John M. Lee: Introduction to topological manifolds (2011) (Chapters 1-3: topological spaces and manifolds, fundamental group, covering spaces).

Group Theory:

• Lorenz Halbeisen: Group Theory, Lecture notes.

Tensor products:

• Kees Dullemond, Kasper Peeters (2010): Introduction to Tensor Calculus.
• Wikipedia: Tensor product (About the first 2/3 of the article until (and including) the section "Linear maps as tensors").
• John M. Lee: Introduction to Smooth Manifolds (2002) (first half of Chapter 12, pp 304-316).
• John M. Lee: Introduction to Riemannian Manifolds (2018) (first half of Appendix B, pp 391-396).

A selection of classic references:

• Manfredo P. Do Carmo: Riemannian Geometry (1992) (classic introduction).
• Gallot, Hulin, Lafontaine: Riemannian Geometry (2004) (advanced and pithy, lots of information).
• Victor Guillemin and Alan Pollack: Differential Topology (1974) Chap 1 (immersions and submersions, Sard's theorem, transversality).
• Wu-Yi Hsiang: Lie Groups (2001) (a beautiful "thin" book) .
• John W. Milnor: Topology from the Differentiable Viewpoint (1965) (differential topology).
• John W. Milnor: Morse Theory (1963) (Part II: thumbnail introduction to connections, curvature, geodesics, Jacobi fields, and the index thm).
• Frank Morgan: Riemannian Geometry: A Beginner's Guide (1992) (intuitive introduction to curvature).
• Michael Spivak: A Comprehensive Introduction to Differential Geometry vols I and vol II (chatty and thorough; vol II, Chap 4 contains an analysis of Riemann's original essay "On the Hypotheses that lie at the Foundations of Geometry").
• J. R. Weeks: The Shape of Space (2019) (intuitive, elementary discussion of spherical, flat, and hyperbolic geometry).

Some pictures of minimal surfaces

• Deformation from helicoid to catenoid
• Scherk Surface
• Costa-Hoffman-Meeks Surface