Differential Geometry I

BMS Course "Differential Geometry I"

Mo 11 a.m.-1 p.m., RUD26, 0'311; Wed 11 a.m.-1 p.m., RUD26, 0'311

Lecturer:           Klaus Mohnke
                           Office: Adlershof, Haus 1, Zimmer 306
                           phone: (030) 2093 1814
                           fax: (030) 2093 2727
                           email:   mohnke@mathematik.hu-berlin.de

Tutorial: Wed 1-3 p.m., RUD 25, 1.115, Alexander Fauck

Office hours: Wed 2 p.m. - 4 p.m., RUD25, 1.306 (office) and by appointment

Homework

Blatt 1
Blatt 2
Blatt 3
Blatt 4
Blatt 5
Blatt 6
Blatt 7
Blatt 8
Blatt 9
Blatt 10
Blatt 11
Blatt 12
Blatt 13

Current and past subjects of the class:

1. Vector Bundles

          fibre bundles (local triviality)
          vector bundles (transition functions, cocycle description, pull-back, direct sum, dual, tensor products)
          covariant derivative (connection form, transition formula)
          exterior derivative, curvature, 2nd Bianchi identity
          principal fibre bundles (Lie groups, fundamental vector fields, (adjoint) representation, associated vector bundle)
          connections and vertical-horizontal decomposition


2. Characteristic classes

Chern forms (closedness and independence)
(elementary) symmetric polynomials
de-Rham-Cohomology (deRham Theorem)
        Chern classes (properties, examples)
        Chern class of the tautological line bundle, Fubini-Study form (hedgehog theorem)

3. Riemannian Geometry

        geodesics as criticak points of the length functional
        exponential map
        Gauss' Lemma
        local minimality of geodesics
        distance metric
        Hopf-Rinow: geodesic complete equals complete

4. Symplectic Geometry

        definition and examples
        Hamiltons equations
        Darboux charts and Weinstein neighborhood theorem
        Lagrangian submanifolds
        compatible almost complex structure

Literature:

Helga Baum: http://www.mathematik.hu-berlin.de/~baum/Skript/diffgeo1.pdf
I.Agricola,Th.Friedrich: Globale Analysis, Vieweg 2001
R.Bott, L.W.Tu: Differential forms in algebraic topology. Springer
A.C.daSilva: Lectures on symplectic geometry, Springer

Klaus Mohnke
Tue, July 9 2013, 2:15 p.m.