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:
  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.