Differential Geometry II
Tue 11:15 a.m.-12:45 p.m.,
RUD26,
0'310; Thu 11:15 a.m.-12:45
p.m., RUD26, 0'310
Please,
register with Moodle for this class.
Lecturer:
Klaus Mohnke
Office: Adlershof, Haus 1, Zimmer 306 absent due to Corona quarantine
phone: (030) 2093 1814 not available
email: mohnke@mathematik.hu-berlin.de
Tutorial: Thu 1:15-2:25 p.m., RUD 25,
3.006
Office hours: Wed 3 p.m. - 4
p.m., ZOOM meeting, information here
Homework (for discussion in tutorials go to the Moodle course of this class)
Set 1
Set 2
Set 3
Set 4
Set 5
Set 6
Set 7
Set 8
Set 9
Set 10
Set 11
Set 12
Tentative schedule:
1. Calculus on manifolds (4/21-
5/7)
some algebra of multi-linear forms (wedge product, interior
product, pull-back, Hodge *) Lit.: 2. and 3., Chapter 1, Slides1
differential forms
(exterior derivative and co-derivative, Poincaré-Lemma), Lit.: 2. and
3, Chapter 2 Slides2
differential forms on
manifolds, Lit.: 2., 3., Chapter 3.3 Slides 3
manifolds with
boundary, orientation, Lit.: 9 (Chapter 1, 13, 14), 10 (Chapter 1) ,
Slides 4
integration of
differential forms, Stokes' Theorem, Lit: 9 (Chapter 14), 10 (Chapter
9) Slides 5
Applications of
Stokes' Theorem, Lit: 2.,3., 10 (Chapter 9) Partition of
Unity Slides 6
2. Vector bundles (5/12-6/11)
fibre
bundle, Lit: 10 (Chapter 6), 7 (Chapter 2) Slides 7
vector bundles: cocycle
description, Lit: 7 (Chapter 2), 8, 9 (Chapter 5), 10 (Chapter 6), 11
(Volume I), 12 (Chapter 3-6) Slides 8
connections: covariant derivatives (dual, direct sum, tensor products), vertical-horizontal decomposition,
connection one-form Lit: 7 (Chapter 3), 8, 10 (Chapter 12), 11 (Violume
I), 12 (Chapter 11-12) Slides 9
space of connections, pull-backs, parallel transport, horizontal
spaces, exterior derivative,
curvature, Lit: 7,8,10,11,12, Slides 10
2nd Bianchi identity, euclidean vector bundles, Slides 11
complex and Hermitian vector bundles, almost complex and Hermitian
structures, Kähler manifolds, 11 (Volume II), 13, Slides 12
frame bundles, Lie groups, principal fibre bundles, Hopf bundle, associated bundles, 12 (Chapter 10-11), Slides 13
associated Lie algebra
bundle, connections, covariant exterior derivatives, curvature,
relation to connections and curvature on vector bundles, 12 (Chapter
10-11), Slides 14
Hopf bundle, quatternionic
Hopf bundle (see also Whiteboard discussion for Problem Set 10),
Chern classes: Axioms,
Chern-Weil construction, Pontrjagin Classes, 11 (Volume II), 12 (Chapter 14), 14 Slides 15 Slides 16
Stiefel Whitney Classes (first and second:
Spin structures), Reductions of principle fibre bundles, 12, 14, Slides 17
3. Calculus of Variations (6/11
- 6/30)
Fundamental Lemma of
Calculus of Variations (see also Slides 17)
Yang-Mills Functional and
Connections, Anti-Self Dual Connections, Gaugfe Theory, 15, 12, Slides 18
Minimal Surfaces and Mean
Curvature, 16,
Lagrangian Mechanics and
Euler-Lagrange Equations. 17, Slides 19
4. Riemannian Geometry (06/30-
07/07)
Geodesics and Jacobi Fields, Slides 20
Conjugated Points and Index Form, Slides 21
Cartan-Hadamard Theorem
Bonnet-Myers Theorem, 18 (Chapter 1), Slides 22
5. Symplectic Geometry (07/09-
07/16)
Literatur: 6
Definition, Examples (Kähler manifiolds, Cotangent Bundle) Slides 23
Hamiltonian Dynamics, Symplectomorphisms, Conservation Laws
isotropic, coisotropic, Lagrangian submanifolds (Examples) Slides 24
Moser's Trick, Darboux' Theorem
Almost complex structures, Holomorphic Curves Slides 25
Symplectic Topology: Exact Lagrangian Embeddings, Non-Squeezing
Literature:
- Helga Baum: http://www.mathematik.hu-berlin.de/~baum/Skript/diffgeo1.pdf
(I will not follow this script but it contains elements of the class)
- I. Agricola,Th. Friedrich: Globale Analysis, Vieweg
2001
(electronic version available in library)
- I. Agricola,Th. Friedrich: Global Analysis, AMS
2002
(English Version, electronic version available via: http://www.ams.org/books/gsm/052/
you have to use HU-VPN!)
- M. Spivak: Calculus on manifolds, Addison-Wesley, New
York, 1965.
- R. Bott, L.W. Tu: Differential forms in algebraic
topology. Springer
- A.C. daSilva: Lectures on symplectic geometry, Springer
(for symplectic geometry if we get there)
- C. Wendl: Lecture notes- Differential Geometry 1, https://www.mathematik.hu-berlin.de/~wendl/Winter2016/DiffGeo1/
- W. Ballmann: Vector Bundles and Connections, http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/conncurv1999.pdf
- J. Lee: Smooth Manifolds, Springer 2003
- J. Lee: Manifolds an Differential Geometry, AMS 2009
- S.Kobayashi, K. Nomizu:: Foundations of Differential
Geometry. Wiley Interscience 1996
- C.H. Taubes: Differential Geometry. Oxford Graduate
Texts in Mathematics, 2011
- A. Moroianu: Lectures on Kähler Geimetry. Cambridge
University Press 2007
- J. Milnor: Characteristic Classes, Princeton
University Press 1974
- D.Freed, K. Uhlenbeck: Instantons and 4- Manifolds,
Springer 1991
- Ch. Bär: Elementary Differential Geometry, Cambridge
UNiversity Press, 2010 (also German version available)
- V.I. Arnold: Mathematical Methods of Classical
Mechanics, Springer 2013
- J.Cheeger, D.G.Ebin: Comparison Theorems in Symplectic Geometry, North-Holland Publishing Company 1975
For
information on HU-VPN see here
(unfortunately only in German)
Klaus Mohnke
Tue, July 22 2020, 7:00 p.m.