# 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

Lecturer
:           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

News:

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 3

Problem Set 4

Problem Set 8

Problem Set 9

Problem Set 10

Problem Set 11

Problem Set 12

Problem Set 13

Problem Set 14

Problem Set 15

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
(01/29)
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

• Helga Baum: http://www.mathematik.hu-berlin.de/~baum/Skript/diffgeo1.pdf (contains stuff we will cover in Differential Geometry II)
• Manfredo do Carmo: Differential Geometry of Curves and Surfaces, several publishers, 2016
• Christian Bär: Elementary Differential Geometry, Cambridge University Press, 2010 (also German version available)
• W. Kühnel: Differentialgeometrie: Kurven - Flächen - Mannigfaltigkeiten. Vieweg-Teubner, 2010
• M. Do Carmo: Riemannian Geometry. Birkhäuser, 1992
• J.M. Lee, Introduction to Smooth Manifolds, Springer
• J. Lafontaine: Introduction to Differential Manifolds, Springer
• John Milnor: Topology from the differentiable viewpoint. Princeton University Press (in particular for the definition of differentialbility on subsets of Rn

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