Geometría Diferencial 2014-2015

Part 1

Date	Content
16/sep/2014	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.
18/sep/2014	Definition of manifold, tangent vector and tangent map. The problem about "avoidance of crossing" is proposed.
23/sep/2014	Calculations with the tangent map referred to the double pendulum. Embedding of a compact manifold in R^D. Some comments about the kinetic energy.
25/sep/2014	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.
02/oct/2014	Interpretation of Lorentz transformations. The cotangent bundle as the phase space. Lagrangian and the map from TM to T*M. Vector mechanics. Conservative force.
07/oct/2014	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.
09/oct/2014	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.
14/oct/2014	Derivation of the first and the second Kepler's laws using Lagrangian mechanics. Integral curves on manifolds.
16/oct/2014	Local and global flows of vector fields. One-parameter groups of diffeomorphisms. Lagrangians invariant by transformations.
21/oct/2014	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.
23/oct/2014	Motivation for Frobenius theorem. Lie bracket. Lie derivative and its geometrical interpretation. Equality between Lie bracket and Lie derivative (naive approach).
28/oct/2014	Proof of the equality between Lie bracket and Lie derivative. Distributions and integral submanifolds. Completely integrable distribution.
30/oct/2014	Proof of the Frobenius theorem
04/nov/2014	Differential forms as fields of alternating forms. Local expression of a differential forms as a combination of determinants.
06/nov/2014	The wedge product as an operation between differential forms. The invariance of the exterior derivative by coordinate changes. Properties of the exterior derivative.
11/nov/2014	Pullback. Integration on manifolds. Partitions of unity. Orientation.
13/nov/2014	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.
02/dec/2014	Scheme of the proof of the Mayer-Vietoris exact sequence. Proof of that cohomology is homotopy invariant.
04/dec/2014	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.
08/jan/2015	Historical introduction: Gauss definition of the curvature. Motivation and meaning of the Riemann tensor.
13/jan/2015	The modern view of the curvature tensor. Geometric meaning of Ricci tensor. Exponential map and Gauss Lemma.
15/jan/2015	Proof of the minimizing property of geodesics using the Gauss lemma. Some results relating curvature and topology

Geometría diferencial (máster) 2014/2015

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