Aarhus University Seal / Aarhus Universitets segl

Mini course by Maciej Czarnecki (University of Lodzki, Poland)

23 – 26 November, 2015, in G3.2 (1532-318)

2015.11.24 | Christine Dilling

Date Mon 23 Nov Thu 26 Nov
Time 14:15    16:00
Location QGM

This is a short description of series of Erasmus+ lectures to be given at
Arhus Universitet, Denmark, on November 23-27, 2015.

Monday + Wednesday: 14:15-15:00
Tuesday + Thursday: 14:15-16:00

These lectures are devoted to study properties of surfaces and foliations from the conformal point of view i.e. via invariants of Mobius transformations.

1 Conformal transformations
We start with a very elementary introduction to conformal transformations
recalling homographies in the complex plane. Then we observe properties of inversions, classify conformal transformations of basic domains, compare with Euclidean case, and explain their role in hyperbolic geometry.

2 Conformal geometry of curves and surfaces
Classical results on the Euclidean theory of curves and surfaces state that some scalar and vector invariants de ne a curve/surface uniquely up to a rigid motion. We identify corresponding quantities invariant under conformal transformations, namely conformal curvature and torsion for curves and conformal principal curvature, their vector elds and the Bryant invariant for surfaces. Examples of typical objects like Dupin cyclides and canal surfaces will be provided.

3 Space of spheres and its applications
We represent codimension 1 oriented spheres in Sn as points in the quadric n+1 in the Lorentz space R1;n+1. We nd interpretation for some families of spheres. Then we see Dupin cyclides and canal surfaces in 4. We also observe constant curvature hypersurfaces in the hyperbolic space Hn.

4 New results in conformal theory of surfaces and foliations
Some of ideas by Langevin, P.Walczak and their collaborators will be presented. Especially we focus on (non)existence of some conformally de ned foliations in model spaces. We present classi cation of canal foliations on S3 and umbilical foliations on Hn.

[1] M. Badura, M. Czarnecki, Recent progress in geometric foliation theory, in Foliations 2012, World Scienti c 2013, 9{21.
[2] R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry, Springer 1992.
[3] A. Bartoszek, P. Walczak, Sz. Walczak, Dupin cyclides osculating circles, Bull. Braz. Math. Soc. 45(1) (2014), 179{195.
[4] G. Cairns, R. Sharpe, L.Webb, Conformal invariants for curves and surfaces in three dimensional space forms, Rocky Mountain J. Math. 24(3) (1994), 933{959.
[5] M. Czarnecki, R. Langevin, Umbilical foliations in hyperbolic spaces, in preparation.
[6] R. Langevin, P. Walczak, Conformal geometry of foliations, Geom. Dedicata 132 (2008), 135{178.
[7] R. Langevin, P. Walczak, Canal foliations of S3, J. Math. Soc. Japan, 64(2) (2012), 659682.
[8] J. O'Hara, Energy of Knots and Conformal Geometry, World Scienti c 2003.