
Mathematics, Faculty of ScienceSigmundur GudmundssonRiemannian GeometrySpring 2020 

Lecturer:
Sigmundur Gudmundsson
Coordinates:
Lecture room 332B, Wednesdays at 13:1516:00
Keywords:
differentable manifolds, submanifolds, tangent spaces, immersions, embeddings,
submersions, tangent bundles, Riemannian manifolds, the
LeviCivita connection, parallelism, geodesics, the curvature tensor,
Jacobi fields and comparison results.
Course Programme: can be found
here.
The course Spring 2019:
Student Survey (Spring 2019) 
Course Report (Autumn 2018)
On the 10th of June 1854
Riemann gave his famous "Habilitationsvortrag" in the Colloquium of the
Philosophical Faculty at Göttingen. His talk with the title
"Ueber die Hypothesen,
welche der Geometrie zu Grunde liegen"
is often said to be the most important in the history
of differential geometry. Riemann's revolutionary ideas generalised the
geometry of surfaces which had been studied earlier by
Gauss,
Bolyai and
Lobachevsky. Later this lead to an exact definition of the concept
of an abstract ndimensional Riemannian manifold. Gauss, at the age of
76, was in the audience and is said to have been very impressed by his
former student.
This course is an introduction to the beautiful theory of
Riemannian Geometry,
a subject with no lack of interesting
examples. They are indeed the key to a good understanding of it and
will therefore play a major role throughout the course. Of special
interest are the classical
Lie groups allowing concrete calculations
of many of the abstract notions on the menu.
Literature: No particular textbook will be used but the
participants are recommended to have a look at some of the following:

M. P. do Carmo,
Riemannian Geometry,
Birkhäuser (1992)

D. Gromoll, W. Klingenberg, W. Meyer,
Riemannsche Geometrie im Grossen,
Lecture Notes in Math. 55,
Springer (1975)

S.Gudmundsson,
An Introduction to Riemannian Geometry,
Lund University (2018)

W. Klingenberg,
Riemannian Geometry,
de Gruyter (1995)

W. Kühnel,
Differential Geometry: Curves  Surfaces  Manifolds,
AMS (2006)

Serge Lang,
Fundamentals of Differential Geometry,
Springer (1999)

John M. Lee,
Riemannian Manifolds,
Springer (1997)

B. O'Neill,
SemiRiemannian Geometry,
Academic Press (1983)

P. Petersen,
Riemannian Geometry,
Springer (2006)

T. Sakai,
Riemannian Geometry,
Translations of Mathematical Monographs 149,
AMS (1996).

M Spivak,
A Comprehensive Introduction to Differential Geometry,
Publish or Perish (1979)
For the history of differential geometry see:

McTutor:
NonEuclidean geometry

D. J. Struik,
Outline of a History of Differential Geometry  I,
Isis, 19 (1933), 92120.

D. J. Struik,
Outline of a History of Differential Geometry  II,
Isis, 20 (1934), 161191.

E. Scholz,
Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis Poincare,
Birkhäuser (1980).
The MacTutor History of Mathematics Archive
Maple routines: curvature.mws