EXTENDED NON-EUCLIDEAN GEOMETRY OBTAINED BY EXTENDING THE GROUP PARAMETERS TO FUNCTIONS OF COORDINATES.
Tsurusaburo Takasu · Institutional Repositories DataBase (IRDB) · 1960
\underline{)}$ $\prime r$ .TAKASU.be noticed that the structure groups are different $f|0/n$ those in their senses, those in the present authour's sense being the extended ones as was explained above.In this paper the extended Non-Euclidean geometries so extended will be established.We have the so-called Cayley-Klein representation (in the classical projective space) and the so-called Poincar\'e-Klein representation (in the Euclidean space and the spherical space) of the classical Non-Euclidean space.These two representations are related to each other by the $so\cdot called$ Darboux-Liebmann transformation ([3], Art.41), which is an extended equiform transformation in the present author's sense.The present extended Non-Euclidean geometries yield us an infinitely many represen- tations of the classical Non-Euclidean geometries, which are gathered together in a single extended $Non\cdot Euclidean$ space, the Darboux-Liebmann transformation being extended. CONTENTS Introduction.\S 4. A Theory of Curves in the Extended Non-Euclidean Geometry.19.Analogues to the Frenet.Serret Formulas in the Extended Non.Euclidean Geometry.$b^{\backslash }X^{\prime}\Gamma ENDED$ NON-ENCLIDEAN GEOMETRY ETC. 3 20.Further Formulas for the Theory of Curves in the Extended Non.Euclidean Geometry.21.A Lemma. 22. Natural Equations of Curves in the Extended Non-Euclidean Geometry.\S 5.A Theory of Hypersurfaces in the Extended Non-Euclidean Geometry.23.Hypersurface $V^{n}$ .24. Curvature Tensor and Carvature Vector.25.Geodesic m-Flat $V^{m}$ and $II\cdot Geodesicm\cdot FlatV^{m}$ immersed in$E^{\prime\iota}*$ .26. Mean Curvature.27.An Extension of the Meusnier's Theorem.28.Asymptotic Curves.29.Lines of Curvature.30.Generalization of the Weingarten's Equations.31.A Generalization of $ccGaussens$ Theorema egregium".32.Fundamental Theorem of the Theory of $V^{m}$ immersed in$E^{n}*$ .33. Ordinary Geodesic Curves referred to II-Geodesic Coordinates.References.\S 1. Preliminaries.