Visualization of Curves and Spheres in Sol Geometry

A. Bölcskei
Department of Descriptive Geometry and Informatics, SzIE YMÉK, Budapest
B.Szilágyi
Department of Geometry, BME, Budapest

Subject: differential geometry, Sol geometry

In [1] W.Thurston based on the ideas about Riemann surfaces formulated a geometrization conjecture for three manifolds which states that every compact orientable three-manifold has a canonical decomposition into a connected sum of ‘simple three-manifolds’, each of which admits one, and only one, of eight homogeneous geometries: H3, S3, E3, S2×R, H2×R, Sol, Nil and SL(2, R). In the past thirty years, many mathematicians have contributed to the understanding of this problem. In 2006 a scoop went round the world claiming that a Russian mathematician, G.Y.Perelman could gave a complete proof of the Thurston conjecture and so the Poincare conjecture, too. Followed by the complex and knotty proof the interest has turned to homogeneous spaces. This talk (based on our earlier works [4], [3]) tries to help in understanding one of the mentioned geometries, the Sol.
The visualization of the three possible two-dimensional homogeneous Riemann geometries E2, H2, S2 is familiar to anyone,  but even in three dimensions, where first time anisotropic cases also appear we have difficulties in imagination. No doubt, the standard models work for E3,H3,S3 and real-time interactive graphics algorithms have been developed by J.R.WEEKS that can be extended even more for the product spaces S2×R, H2×R [2]. The remaining three Thurston`s homogeneous 3-dimensional geometries however are difficult to handle. 

		[1] W.P.Thurston: Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357-381.
		[2] J.F. Weeks: Real-Time Animation in Hyperbolic, Spherical and Product Geometries, Non-Euclidean Geometries (edited by A. Prékopa and E. Molnár) Mathematics and Its Applications 581 Springer, (2006), 287-306.
		[3] E. Molnar- B. Szilagyi: Translation curves and their spheres in homogeneous geometries. (manuscript)
		[4] A. Bölcskei - B. Szilágyi: Frenet Formulas and Geodesics in Sol Geometry, (to appear in Beiträge zur Algebra und Geometrie)