Geometría diferencial (máster)
Differential Geometry 2015/2016



Recent updates:
01/nov/2015 Information about the composition has been posted here.
3/nov/2015 The problem set 2 is available.
5/nov/2015 Solutions of the previous non-mandatory problems and statement of a new one.
12/nov/2015 A new non-mandatory problem.
17/nov/2015 Yet another non-mandatory problem.
26/nov/2015 More comments about the composition here.
Important: If somebody is willing to attend the exam instead of handling the composition, please let me know in advances because I have to teach on Thurday 21.
No lecture on Tuesday 19. To deliver problem set 3, use email or leave a copy in Secretaría de Matemáticas to be put in my mailbox.



Course details

The contents, according to the guía docente employed last year is:
  1. Introduction. Basic differential geometry. Vectors and tensors. Vector fields and flows. Lie bracket and Frobenius theorem.
  2. Differential topology. Differential forms. Stokes theorem. De Rham Cohomology.
  3. Riemannian geometry. Mechanics, metrics and geodesics. Levi-Civita connection and tensor calculus. Curvature. Introduction to general relativity.
The experience of the last course suggest to reduce the contents of the second chapter and devote more time to Riemannian geometry. With this idea, we shall try to enter into the topics of Riemannian geometry shortly after the beginning of the course. In connection with this, some reordering of the contents listed before is expected as well as a fine (or not so fine) tunning according with the student background.
In case you miss a lecture a brief summary of each of them wil be posted here.

It is intended to use mechanics and other topics on Physics as motivation and to provide examples and applications. No prior knowledge on Physics is assumed beyond the very basic concepts.

The lectures are scheduled on Tuesday and Thrusday 4:00pm - 5:30pm in room 320 (Building 17, Science Faculty, Mathematics).

For general reference, please check the master program website in the Department of Mathematics.




In this course we follow mainly the notes in Spanish Geometría Diferencial. They do not cover De Rham cohomology that is included Geometría IV 2008-2009. Another main source is the book by do Carmo on Riemannian geometry.
Occasionally some files will be posted in the table below, with summaries of some topics.

File Status
old_notes1 (to be updated)
old_notes2 (to be updated)

Corrections and comments are wellcome.




The following sheets are home assignments that contribute to the grading.

Home assignments
File Status Deadline
Problem set 1
Available 3/nov/2015
Problem set 2
Available 10/dec/2015
Problem set 3
Available 19/jan/2015

The grades for officially enrolled students can be checked in the Moodle server after the home assignments are corrected.

From time to time some non mandatory exercises will be proposed.

Non mandatory exercises

File Deadline Solution
Compute dimension
Induced metric on S^2
nms02 +0.1
Minimal surface collapse
15/oct/2015 nms03 +0.1
Stereographic metric on Sn

A conservation law 19/nov/2015
nms05 +0.3
2d curvature
nms06 +0.75
5) in problem set 3

7) in problem set 3 19/jan/2016




The grading is as follows:

1) Home assignments 50%.
2) Final exam or extra activities 30%.
3) Quizzes, in-class exercises, participation 20%

More details will be given along the course.
"Extra activities" in 2) means a brief composition about a topic related to the course.

About the composition: It is a short essay of 10-15 typed pages. The idea is to give an overview about a topic not covered in the course. You are not expected to give fine details. You should emphasize the general ideas showing that you know what is the topic about. The deadline is 14th January 2016.

(Added November 26th) The main guideline for the composition is to emphasize the ideas. You should try to show that you have understood the motivation of the topic, the main ideas, the achivements, the targets, the conjectures. Please do not try to be very technical. I'd prefer a composition addressed to an educated layman reader. A couple of you have chose topics so far from my knowledge that this is close to reality.
Roughly speaking, you should write theorems but do not forget to show me that you have learned something beyond the scheme lemma-theorem-proof that I can find in any book. Originality is a crucial point and it will be considered in the assessment.

You can choose freely any topic related to the course (even loosely. If it is the case, ask for confirmation). The following list will be updated from time to time.

Suggested topics for the composition
Lie groups and algebras in differential geometry
Gauss-Bonnet theorem in higher dimensions
Symplectic geometry and mechanics
Fiber bundles and gauge theories
The sphere theorem (in Riemannian geometry)
Whitney immersion theorem

Some of you have asked about bibliography. I include here something but, please, do not follow it blindly because my knowledge is very limited in some of the topics and finding the good references is part of your work. Your main task is not understanding a topic on a book but using different books and papers (including the one you understand) to give an overview.

1) Warner. "Foundations of differentiable manifolds and Lie groups".
2) S.-S. Chern, "On the curvatura integra in Riemannian manifold", Annals of Mathematics 46 (4): 674-684. (Understanding the full proof can be difficult. It is enough for the composition to follow a sketch of the ideas and to understand the elements appearing in the statement and its relation with the 2 dimensional case).
M. Taylor. Partial differential equations. Volume II Appendix C.
C. Zhu. The Gauss-Bonnet theorem and its applications.
There are some brief comments in Berger. "A panoramic view of Riemannian geometry"
3) Arnold. "Mathematical methods of classical mechanics".
5) Do Carmo. "Riemannian geometry".
Cheeger & Ebin. "Comparison theorems in differential geometry".
6) Gualteri.
Hirsch. "Differential topology".
It is worthy to look Whitney's original papers.

Other topics that you have considered for your composition:

Gravitational waves:
Schutz. "A first course in general relativity".
Misner, Thorne & Wheeler. "Gravitation" (this is a very thick book but it is not an encyclopedia. The explanations are good and it is readable in general).

Bott & Tu. Differential forms in algebraic topology.

Lie algebras and physics:
H. Georgi. "Lie algebras in particle physics from isospin to unified theories".
M. Hammermesh "Group Theory and its Application to Physical Problems"
These can be hard for a first encounter with physics. Just understanding the relation between SU(2), SO(3), spin and angular momentum in quantum mechanics can be the topic for a composition if you do not know too much about physics. It appears in many of the books on quantum mechanics.



For this subject:


Content of the lectures

Part 1
Plan of the course. Anonymous survey to sound out the level and interest of the students. Some comments about the motivation of the definition of manifold. The double pendulum as an example.
17/sep/2015 Basic definitions in differential geometry. Coordinate charts. Naive computation of the dimension of some manifolds defined by matrices: SL_n,GL_n, etc.
22/sep/2015 Tangent vectors. Covectors. Tangent map (pushforward). Einstein summation convention. Tensors as multilinear forms.
24/sep/2015 Tensors as multilinear forms. Components of a tensor.
29/sep/2015 Tensor in manifold. Vector fields. Tangent bundle. Cotangent bundle. The transformation rule for tensors.
01/oct/2015 No lecture
Examples of tensors and their components. Lorentz transformations as linear map preserving Minkowski metric.
Pullback. Induced metric. The basic result in calculus of variations. Conservation of the energy. An example of minimal surface.
Proof of the results of caluclus of variations. Some basic ideas about classical mechanics.
Kinetic energy and potential energy. Examples of Lagrangian mechanics: the simple pendulum and the doble pendulum.
Noether's theorem. Conservation laws. Planetary motion.
Riemannian and semiriemannian manifolds. Mechanical definition of geodesics in R^3. Computation of geodesic equations through the Lagrangian. Examples.
The differential equations of geodesics on the sphere. General form of the defining equations of the geodesics. Christoffel symbols.
Basic ideas in general relativity. Example: simulating gravitation with a metric.
Heuristics about metrics simulating Newtonian gravitation. The Schwarzschild metric. Black holes.
Radial temporal geodesics of the Schwarzschild metric and its interpretation as free falling particules. The role of the singularity at the "event horizon".
Radial lightlike geodesics in the Schwarzschild geometry. Connections: motivation and mathematical definition.
Connections and gauge transformations. Existence and uniqueness of the Levi-Civita connection.
Three approaches to curvature: PDEs, tensorial calculus, geometry. The case of dimension 2. An algorithm for Gaussian curvature.
Proof of the well-definedness of Gaussian curvature and K=0 if and only if the metric is euclidean in a coordinate system. Local isometries.
Analysis of the PDE showing the non-flatness of the sphere. Definition of the Riemann tensor. Re-definition through an operator.
Proof of the tensor properties of the curvature tensor. Curvature tensor=0 and local flatness are equivalent. First steps in the proof. A result about differential equations (the existence of a gradient function revisited).
Curvature tensor=0 and local flatness are equivalent: end of the proof. Integral curves and flows. Generalization to higher dimensions: Frobenius theorem.
Examples on Frobenius theorem. Geometric idea. Curvature and deviation of geodesics. The equation for geodesic deviation.
Proof of the geodesic deviation equation. Some ideas around the field equations in general relativity.
The field equations and the geodesic deviation. The Hilbert action.
Basic facts in topology. Alternate forms.
Exterior product. Differential forms on manifolds.
Pullback of a form. Integration of differential forms.
Manifolds with boundary. Stokes theorem. An application to topology.



Configuration space
Degrees of freedom
Submanifold equations
Generalized coordinates
Coordinate map
Tangent vector
Velocity phase space
Tangent bundle
Phase space
Cotangent bundle
Lagrangian L
Function L:TM->R
Generalized momentum p ∂L/∂q̇‌·