Abstract
The motivation for this talk and paper is related to the classification of the homogeneous simply connected maximal 3-geometries (the so-called Thurston geometries: E 3 , S 3 , H 3 , S 2 × R , H 2 × R , S L 2 R ˜ , Nil , and Sol ) and their applications in crystallography. The first author found in (Molnár 1997) (see also the more popular (Molnár et al. 2010; 2015) with co-author colleagues, together with more details) a unified projective interpretation for them in the sense of Felix Klein’s Erlangen Program: namely, each S of the above space geometries and its isometry group Isom ( S ) can be considered as a subspace of the projective 3-sphere: S ⊂ P S 3 , where a special maximal group G = Isom ( S ) ⊆ Coll ( P S 3 ) of collineations acts, leaving the above subspace S invariant. Vice-versa, we can start with the projective geometry, namely with the classification of Coll ( P S 3 ) through linear transforms of dual pairs of real 4-vector spaces ( V 4 , V 4 , R , ∼ ) = P S 3 (up to positive real multiplicative equivalence ∼) via Jordan normal forms. Then, we look for projective groups with 3 parameters, and with appropriate properties for convenient geometries described above and in this paper.
Highlights
Our intention is to investigate and visualize the possible projective transforms, not considered earlier, for 3-parameter transitive translations, the possible invariant projective polarities, the possible invariant Riemann metrics, etc
We conjecture that our experience space can wear the structure of these 8 Thurston geometries, in small size at certain physical circumstances
The 4th Sol geometry above is an affine metric space in A3 with strange metrics [7]; we indicate g another Sol interpretation as well
Summary
Our intention is to investigate and visualize the possible projective transforms, not considered earlier, for 3-parameter transitive translations, the possible invariant projective polarities, the possible invariant Riemann metrics, etc. The ideal plane e0 is opposite to the origin, and contains the ideal points E1 , E2 , E3 of the x, y, z axes, respectively We extend this translation group with all projective collineations, leaving invariant a projective polarity (or scalar product) of signature (+, +, +, 0), with unimodular linear transforms, as usual. We have, in each an infinitesimal (positive definite) Riemann metric, invariant under certain translations, guaranteing homogeneity in every point These translations commute only in E3 , in general, but a discrete (discontinuous) translation group—as a lattice—can be defined with compact fundamental domain in Euclidean analogy, but with some different properties. A Gum-fibre model of Hans Havlicek and Rolf Riesinger, used by Hellmuth Stachel with other respects (Vienna UT)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
