## Mathematics, Faculty of Science## Sigmundur Gudmundsson## Differential Geometry## Autumn 2015 |

**Lecturer:** Sigmundur Gudmundsson

**Coordinates:** Lecture room 332B

Around 300 BC Euclid wrote "The Thirteen Books of the Elements". It was used as the basic text on geometry throughout the Western world for about 2000 years. Euclidean geometry is the theory one yields by assuming Euclid's five axioms, including the parallel postulate.

Gaussian geometry is the study of curves and surfaces in three dimensional Euclidean space. This theory was initiated by the ingenious Carl Friedrich Gauss (1777-1855). The work of Gauss, János Bolyai (1802-1860) and Nikolai Ivanovich Lobachevsky (1792-1856) lead to their independent discovery of non-Euclidean geometry. This solved the best known mathematical problem ever and proved that the parallel postulate was indeed independent of the other four axioms that Euclid used for his theory.

We show that a curve in $$**R**^{3} is, up to
Euclidean motions, totally determined by its curvature and torsion.
We study the second fundamental form of a surface, describing its
shape in the ambient space $$**R**^{3}.
This leads to a fundamental object the curvature of the surface.
Amongst many interesting results we prove the remarkable
"Theorema Egregium"
of Gauss which tells us that the curvature is an intrinsic object i.e.
determined by the way we measure distances on the surface.
We study geodesics which locally are the shortest paths connecting
points on the surface.
Furthermore we prove the astonishing Gauss-Bonnet theorem.
This implies that for a compact surface the curvature integrated over
it is a topological invariant.

**Literature:**

A. Pressley, *Elementary Differential Geometry* (2nd Edition), Springer (2010)

S. Gudmundsson, *An Introduction to Gaussian Geometry*, Lund University (2015)

The Gaussian geometry treated in this course is a requisite for the still active areas of Riemannian geometry and Lorentzian geometry. The latter is the mathematical basis for Einstein's theory of general relativity.

**Earlier written exams:**

28 October 2015, 14 November 2015

University of Indiana: Minimal Surface Archive

For the history of differential geometry see:

MacTutor: Non-Euclidean geometry

D. J. Struik,
*Outline of a History of Differential Geometry - I*,
Isis, **19** (1933), 92-120

D. J. Struik,
*Outline of a History of Differential Geometry - II*,
Isis, **20** (1934), 161-191

E. Scholz,
*Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis Poincare*,
Birkhäuser (1980)