Differential Geometry I

BMS Course "Differential Geometry I"

  Wed 11:15 a.m.-12:45 p.m., RUD25, 1.115; Thu 3:15 p.m.-4:45 p.m., RUD25, 1.013

:           Klaus Mohnke
                           Office: Adlershof, Haus 1,  Zimmer 306
                           phone: (030) 2093 1814
                           fax: (030) 2093 2727
                           email: mohnke at math dot hu-berlin dot de

Tutorial:  Wed 9:30 -11 p.m., RUD 25, 1.115,  Mihai Munteanu      

Office hours: Wed 2 p.m. - 3 p.m., RUD25, 1.306 (office) and by appointment


The exams on March 30/31 will be postponed to a later yet to be fixed date. You will
be notified via email or will be informed on this webpage.

Exams will be oral exams (30 minutes). 

           A   list of sample exam questions  updated on  02/21 at 1:55 p.m.!!

           A skript for Gauss-Lemma and the Theorem of Hopf Rinow

Homework problems

Warm Up

Problem Set 1

Problem Set 2

Problem Set 3

Problem Set 4

Problem Set 5 

Problem Set 6

Problem Set 7

Problem Set 8

Problem Set 9

Problem Set 10

Problem Set 11

Problem Set 12

Problem Set 13

Problem Set 14

Problem Set 15

Notes and comments by Mihai Munteanu

Prospective subjects of  class:

1. Curves and Surfaces in Euclidean space (10/16 - 12/18)

          regular curves in Rn: length parametrization (10/16)
          plane curves, curvature, Frenet's formula, turning number, simple closed curves (10/17 - 10/31)
          space curves, curvature, torsion, Frenet's formula, total curvature and total angle, Fáry-Milnor Theorem
(11/06 - 11/20)
          surfaces in R3: regular parametrizations (immersions), normal field, orientation (11/21- 11/28)
          first and second fundamental form, principal curvatures, mean curvature, Gaussian curvature (12/04- 12/11)
          Gaussian curvature of compact surface is positive somewhere, computations of curvature, geometric
          interpretation of curvature,  ruled surfaces, convexity (12/12-12/18)

2. Riemannian Geometry (12/19 - 02/13)

manifolds: topological notation and definition, submanifolds of Rn, examples (12/19 - 01/08)
          tangent vectors and cotangent vectors, tangent bundle and cotangent bundle (01/09)
          differentiable maps, push-forward and pull-back (01/09)
          differentiable vector fields and 1-forms, Lie-bracket (01/09)
          vector fields along maps (01/15)
  Riemannian metrics, examples, distance, metric (01/15)
          geodesics, geodesic equation, Christoffel symbols (01/16)
          covariant derivatives, Levi-Civita connection (01/22)
          curvature tensor, Riemann curvature (01/23)
          tensors, Bianchi identities (01/23)
          Ricci and scalar curvature, sectional curvature, Riemann curvature of surfaces
          Gauss' Theorema Egregium (01/30)
          Euler characteristic (tentativly), Poincaré-Hopf-Theorem, Gauss-Bonnet Theorem (02/05-02/06)
          exponential map and normal coordinates, Gauss-Lemma (02/06-02/12)
          Hopf-Rinow: geodesic completeness equals metric completeness (here is  a skript ) (02/13)


Literature:  books in which I looked, but no particular recommendation for buying

Klaus Mohnke
Thu, March  19  2020, 2:00 p.m.