- Authors
- Emad N. Naseem Shonoda
- title
- On Ruled Surfaces in three-dimensional Minkowski Space
- Please use the following URL when quoting:
- https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa-63555
- Date of submission
- 16.07.2010
- Date of defense
- 13.12.2010
- Abstract (EN)
- In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common
- Keywords (DE)
- Regelflächen, Sphärische Bild, Kruppa\'s Invarianten, Kruppa-Sannia Begleitbeins, Strikitionlinie, Minkowski Raum, Birkhoff Orthogonalität, Semi-inneres Produkt,
- Keywords (EN)
- Ruled surfaces, spherical image, Kruppa’s differential invariants, Kruppa-Sannia moving frame, striction curve; Minkowski space, Birkhoff orthogonality, semi-inner product
- Classification (DDC)
- 510
- Classification (RVK)
- SK 370
- Examiner
- Prof. Dr. Gunter Weiss
- Prof. Dr. Horst Martini

- Supervisor
- Prof. Dr. Gunter Weiss

- Awarding institution
- Technische Universität Dresden, Dresden
- URN Qucosa
- urn:nbn:de:bsz:14-qucosa-63555
- Qucosa date of publication
- 22.12.2010
- Document type
- doctoral_thesis
- Document language
- English
- licence