Thèse--Nancy.

Thesis--Rijksuniversiteit te Leiden, 1936.

Thesis--Rijksuniversiteit te Leiden, 1936.

Thesis--Friedrich-Wilhelms Universität, 1937. / Lebenslauf.

Thesis--Rijks-universiteit Leiden, 1915.

In the study of advanced geometry, we shall deal with a certain important relation between pairs of figures in space, and also between their properties. There are two distinct parts to analytic geometry, the analytic work and the geometric interpretation. Two systems of geometry, depending upon different elements with the same number of coordinates, will have the same analytic expressions and will differ only in the interpretation of the analysis. In such a case it is often sufficient to know the meaning of the coordination and the interpretation of a few fundamental relations in each system in order to find for a theorem in one geometry a corresponding theorem in the other. The nature of this relation is explained by the theorem of duality which assets that a dual, or reciprocal, statement can be derived from a given statement.

No description available.

No description available.

The study of deformation zones is of great geological interest since these zones can separate rocks with different characteristics. The geometry of these structures with depth is important for interpreting the geological history of an area. Paper I to III present 2D reflection seismic data over deformation zones targeting structures in the upper 3-4 km of the crust. These seismic profiles were acquired with a crooked-line recording geometry. 2D seismic processing assumes a straight recording geometry. Most seismic processing tools were developed for sub-horizontally layered structures. However, in the crystalline rocks in Scandinavia more complex structures with contrasting dip directions and folding are common. The crooked-line recording geometries have the benefit of sampling a 3D volume. This broader sampling can be used to gain knowledge about the true geometry of subsurface structures. Correlation with geological maps and other geophysical data along with seismic data modeling can be used to differentiate reflections from faults or fracture zones from other reflectivity, e.g. mafic bodies. Fault and fracture zones may have a large impedance contrast to surrounding rocks, while ductile shear zones usually do not. The ductile shear zones can instead be interpreted based on differing reflectivity patterns between domains and correlations with geology or magnetic maps. Paper IV presents 3D reflection seismic data from a quick-clay landslide site in southern Sweden. The area is located in a deformation zone and structures in unconsolidated sediments may have been influenced by faults in the bedrock. The main target layer is located at only 20 m depth, but good surface conditions during acquisition and careful processing enabled a clear seismic image of this shallow layer to be obtained.The research presented in this thesis provides increased knowledge about subsurface structures in four geologically important areas. The unconventional processing methods used are recommended to future researchers working with data from crooked-line recording geometries in crystalline environments. The imaging of shallow structures at the quick-clay landslide site shows that the 3D reflection seismic method can be used as a complement to other geophysical measurements for shallow landslide site investigations.

In der vorliegenden Arbeit werden Geradenkongruenzen des projektiv abgeschlossenen dreidimensionalen euklidischen Raumes differentialgeometrisch untersucht. Nach J. PLÜCKER lassen sich Geraden in gleicher Weise als Grundelemente eines Geradenraumes auffassen wie die Punkte in einem Punktraum. Unter Beachtung dieser Überlegung scheint eine "natürliche" Behandlung der Geradenkongruenzen interessant und sinnvoll. Sie bildet den Gegenstand der vorliegenden Dissertation. Ein besonderes Augenmerk richtet sich dabei auf die Frage nach "kleinsten" Geradenkongruenzen ("Minimalkongruenzen") in der Geradenmenge des reellen projektiv abgeschlossenen dreidimensionalen euklidischen Raumes. Dahinter verbirgt sich eine gewisse Analogiebildung in der Liniengeometrie, die der klassischen Differentialgeometrie entstammt. Die Geradenkongruenzen bilden hierbei das liniengeometrische Analogon zu den Flächen des dreidimensionalen (Punkt-)Raumes. Das Wort "Kleinste" stellt im Geradenraum einen Bezug zu den Minimalflächen in der Differentialgeometrie her. Nun gestatten diese Fragestellungen in der Liniengeometrie eine anschauliche Interpretation, sobald man ein Punktmodell des Geradenraumes vorliegen hat. Einparametrige Geradenmannigfaltigkeiten (Regelflächen) lassen sich darin als Kurven und Geradenkongruenzen als zweidimensionale Flächen auffassen. Die vierparametrige Geradenmenge des reellen projektiven dreidimensionalen Raumes ist in diesem Modell eine Quadrik vom Index 2 in einem reellen projektiven fünfdimensionalen Raum, die so genannte KLEINsche Hyperquadrik. Der Modellwechsel wird durch die KLEINsche Abbildung vollzogen. / In the available work line congruences of the projectively extended three-dimensional euclidean space will be analysed. Following to J. PLÜCKER lines can be seen as basic elements of an line space like in the same way points in a point-space. Taking this fact in consideration a "natural" handling with line congruences might be interesting and reasonable. A special detail in the thesis is the question to minimal congruences in the set of lines of the projectively extended euclidean three-space. It can also be seen as an analogous problem in the geometry of lines which can be find in the differential geometry of surfaces. In this case the line congruences are similar to the surfaces of the three-dimensional (point-)space. The phrase "minimal" means in the line space the connection to the minimal surfaces in the differential geometry. These questions offer in line geometry demonstrative interpretation possibilities if a point-model in the line space exists. One-parameter manifolds of lines (rule surfaces) can be seen in this ambiance as curves and line congruences as two dimensional surfaces. The four-parametric set of lines in the projectively extended three-dimensional euclidian space is in this model a quadric of the index 2 in a real projective five-dimensional space, the so called KLEIN-quadric. The changing of the model is managed by the KLEIN-mapping.