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 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 (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.