# 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)

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
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:

1. 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)
2. I. Agricola,Th. Friedrich: Globale Analysis, Vieweg 2001 (electronic version available in library)
3. 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!)
4. M. Spivak: Calculus on manifolds, Addison-Wesley, New York, 1965.
5. R. Bott, L.W. Tu: Differential forms in algebraic topology. Springer
6. A.C. daSilva: Lectures on symplectic geometry, Springer (for symplectic geometry if we get there)
7. C. Wendl: Lecture notes- Differential Geometry 1, https://www.mathematik.hu-berlin.de/~wendl/Winter2016/DiffGeo1/
8. W. Ballmann: Vector Bundles and Connections, http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/conncurv1999.pdf
9. J. Lee: Smooth Manifolds, Springer 2003
10. J. Lee: Manifolds an Differential Geometry, AMS 2009
11. S.Kobayashi, K. Nomizu:: Foundations of Differential Geometry. Wiley Interscience 1996
12. C.H. Taubes: Differential Geometry. Oxford Graduate Texts in Mathematics, 2011
13. A. Moroianu: Lectures on Kähler Geimetry. Cambridge University Press 2007
14. J. Milnor: Characteristic Classes, Princeton University Press 1974
15. D.Freed, K. Uhlenbeck: Instantons and 4- Manifolds, Springer 1991
16. Ch. Bär: Elementary Differential Geometry, Cambridge UNiversity Press, 2010 (also German version available)
17. V.I. Arnold: Mathematical Methods of Classical Mechanics, Springer 2013
18. 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.