Geometría diferencial (máster) 2014/2015



Esta página se actualizará continuamente a lo largo del curso.

broken_english This web page is not fully translated but following this icon, in each part there are some comments in (likely broken) English.  The topics covered in each lecture are listed here.

broken_english Here you can find a brief approximate dictionary Mechanics-Geometry.

broken_english  Important dates:
January 15: deadline for the composition and last lecture.
January 20: deadline for Sheet 3 and final exam (non-mandatory).




El temario previsto, tomado de la guía docente es el siguiente:
  1. Introducción. Resumen de geometría diferential elemental. Vectores y tensores. Campos de vectores y sus flujos. Corchete de Lie y el teorema de Frobenius.
  2. Topología diferencial. Integración en variedades. Formas diferenciales. El teorema de Stokes en variedades. Cohomología de De Rham.
  3. Geometría riemanniana. Mecánica, métricas y geodésicas. Conexión de Levi-Civita y cálculo tensorial. Curvatura en variedades. Introducción a la relatividad general.
En cada uno de estos capítulos se prevé utilizar la mecánica y algunos otros temas de física como ejemplo, motivación y aplicación.

Más detalles acerca del curso se pueden encontrar en la guía docente.

El horario de las clases es de 16:00 a 17:30 martes y jueves en el aula 320 del módulo 17.

Hay una página en el Departamento de Matemáticas para referencia general de las asignaturas de Máster.

broken_english There is an English translation of the course contents in the guía docente if you need it. In each part of the course it is intended to use mechanics and other topics on Physics as motivation and to provide examples and applications. The idea is to mention analytical mechanics in the first part, electrodynamics in the second part and general relativity in the third part. No prior knowledge on Physics is assumed. The lectures are scheduled on Tuesday and Thrusday 16:00-17:30 in room 320 (Building 17, Science Faculty, Mathematics).]




El curso está en cierta medida reflejado en los apuntes Geometría Diferencial, con la salvedad de que allí se tratan también temas de geometría global y no se cubre la cohomología de De Rham, que sí está en Geometría IV 2008-2009. Los siguientes ficheros tratan algunos temas que están de alguna forma relacionados con la Física, especialmente con la mecánica. Son parte del curso. Prácticamente no requieren conocimientos previos no matemáticos.

Fichero Estado
fisgeo2.pdf Disponible
fisgeo3.pdf Disponible
fisgeo4.pdf No disponible

Agradezco que se me comuniquen las posibles erratas.

broken_english There are lecture notes in Spanish Geometría Diferencial. They do not cover De Rham cohomology that perhaps will be studied using Geometría IV 2008-2009. In the previous table there are some topics related to Physics. Again, no prior contact with Physics is assumed. A short bibliography in English was mentioned the first day. A long list is in the guía docente. My guess (I am not sure about it) is that mathematical written Spanish should not be impossible to follow with a reasonable training for western European students.




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

Home assignments
File State Deadline
Sheet1.pdf Available October 30
Sheet2.pdf Available December 11
Sheet3.pdf Available January 20

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 as extra activities.

Non-mandatory exercises
Topic Deadline Solution
Eigenvalues and dimension September 30
Euler-Lagrange by hand October 14 sol
Definition of the exterior derivative November 13 sol

Here you can read an extended version of the first non-mandatory exercise.




La calificación final provendrá de tres tipos de pruebas cuya contribución se indica a continuación:

1) Hojas de problemas: 50%.
2) Examen final o actividades extra: 30%.
3) Pequeños controles y participación en clase: 20%

Más detalles acerca de las pruebas se darán en el curso.
Las "actividades extra" del punto 2) se refieren a un pequeño trabajo. La temática es libre dentro de límites razonables. Se anunciarán las restricciones con respecto al formato.

broken_english 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. The topic can be chosen freely up to reasonable restrictions. The limitations regarding the format will be announced.

More details about grading:
Naming H, E and C the grades corresponding to each part, the final mark is
F = 0.5*H+0.3*E+0.2*C.

H is (H1+H2+H3)/3 where Hn is the grade of the sheet n.

E is max(Final exam, Composition). Note that neither the exam nor the composition is mandatory. The final examination is scheduled for January 20th. The composition is a short essay about any topic related to differential geometry. I recommend to ask me for confirmation after choosing the topic. The advisable length is 10-15 pages (double-spaced, 11pt or 12pt). The deadline is January 15th 2015. [The idea is choosing between Final exam and Composition but it is allowed to do both].

C You can get the full grade in this part attending regularly the lectures and solving at least  a non-mandatory exercise. Some other sources of participation can be also counted.

The contribution of other activities (e.g. non-mandatory exercises beyond the one counted in C) will be added to the final grade.



De la asignatura:


Content of the lectures

Part 1
Plan of the course. Anonymous survey to sound out the level and interest of the students (this one). A brief dictionary between geometry and mechanics through the double pendulum example.
Definition of manifold, tangent vector and tangent map. The problem about "avoidance of crossing" is proposed.
Calculations with the tangent map referred to the double pendulum. Embedding of a compact manifold in R^D. Some comments about the kinetic energy.
Kinetic energy in generalized coordinates for the double pendulum. (Euclidean) Tensors, Einstein summation convention. The tangent bundle.
30/sep/2014 The cotangent bundle. Tensors on a mainfold. Transformation law for tensors. Examples. Minkowski metric tensor.
Interpretation of Lorentz transformations. The cotangent bundle as the phase space. Lagrangian and the map from TM to T*M. Vector mechanics. Conservative force.
Euler-Lagrange equations. Equivalence with the Newton's second law for free particles. Calculus of variations. Conservation of the energy. The simple pendulum using vector mechanics and Lagrangian mechanics.
Conservation of the energy and other observations with regards to the simple pendulum. Examples of applications to the geometry of surfaces: geodesics and minimal surfaces. The vector mechanics formulation of the planetary motion.
Derivation of the first and the second Kepler's laws using  Lagrangian mechanics. Integral curves on manifolds.
Local and global flows of vector fields. One-parameter groups of diffeomorphisms. Lagrangians invariant by transformations.
Noether's theorem. Conservation of the linear momentum for interactions depending on the distance. Conservation of the angular momentum for interactions with spherical symmetry. Examples of flow and group law.
Motivation for Frobenius theorem. Lie bracket. Lie derivative and its geometrical interpretation. Equality between Lie bracket and Lie derivative (naive approach).
Proof of the equality between Lie bracket and Lie derivative. Distributions and integral submanifolds. Completely integrable distribution.
Proof of the Frobenius theorem
Differential forms as fields of alternating forms. Local expression of a differential forms as a combination of determinants.
The wedge product as an operation between differential forms. The invariance of the exterior derivative by coordinate changes. Properties of the exterior derivative.
Pullback. Integration on manifolds. Partitions of unity. Orientation.
Definition of manifold with boundary. Some topological considerations. Statement of the Stokes theorem. The classical examples.
18/nov/2014 The field x/|x|^3 and differential topology. Maxwell equations and the Stokes theorem. Correction of the Ampère law with the charge conservation law.
20/nov/2014 Historical comments about the fundamental group, homology and cohomology. Definition of the de Rham cohomology. H^0 and the connectness.
25/nov/2014 The pullback acting on H^k and its relation with the homotopy. Poincaré's Lemma. The group H^n is isomorphic to R via integration for an orientable compact connected n-dimensional manifold.
27/nov/2014 Cohomology ans differential topology: Brouwer fixed-point theorem. Exact sequences. The Mayer-Vietoris sequence. The cohonology groups of the spheres. The cohomology groups of the torus.
Scheme of the proof of the Mayer-Vietoris exact sequence. Proof of that cohomology is homotopy invariant.
The hairy ball theorem and other topological results. Underlying ideas in Riemannian geometry.
09/dec/2014 Motivation and notation for the concept of metric. Mechanical interpretation of the geodesics. Examples. The Schwarzschild metric.
11/dec/2014 Computation of a geodesic for the Schwarzschild metric. Newtonian approximation for weak fields. Motivation for the definition of connection.
16/dec/2014 Definition of connection. An analogy: Coriolis acceleration. Covariant derivative. The Levi-Civita connection.
18/dec/2014 Christoffel symbols. Parallel transport "absolute parallelism".  Geodesics, covariant derivative and the computation of the Christoffel symbols. An example. Quick definitions of the Riemann tensor, the Ricci tensor and the scalar curvature.
Historical introduction: Gauss definition of the curvature. Motivation and meaning of the Riemann tensor.
The modern view of the curvature tensor. Geometric meaning of Ricci tensor. Exponential map and Gauss Lemma.
Proof of the minimizing property of geodesics using the Gauss lemma. Some results relating curvature and 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/∂\dot{q} ‌·