# Conformal geometry of surfaces with applications to foliations

Maciej Czarnecki
(University of Lodzki, Poland)
Minikursus
23 – 26 november, 2015, in G3.2 (1532-318)
Abstrakt:

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

Introduction

These lectures are devoted to study properties of surfaces and foliations from the conformal point of view i.e. via invariants of Møbius 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 define 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 $S^n$ as points in the quadric $n+1$ in the Lorentz space $R^{1;n+1}$. We find 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 $H^n$.

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 defined foliations in model spaces. We present classification of canal foliations on S3 and umbilical foliations on $H^n$.

References

[1] M. Badura, M. Czarnecki, Recent progress in geometric foliation theory, in Foliations 2012, World Scientific 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 Scientific 2003.

Organiseret af: QGM
Kontaktperson: Andrew Swann