How come Gauss beside Bólyai and Lobachewski THEOREMA EGREGIUM How come Gauss beside Bólyai and Lobachewski © 2017 Dr. Prof. Peter G. Gyarmati BGL-2017, Gyöngyös

Metric space Metric space: the distance of any two points measurable:   This postulate establish the metric geometry – proved by Gauss   Infinitesimally: ( ; etc.) Conditions: valid in „small” environment, continuous & differentiable Gauss coordinates: x=x(u,v); y=y(u,v); z=z(u,v) (x, y, z) external coordinates (u, v) surface coordinates Infinitesimally: ; ; Gaussian coordinates dr. prof. P. G. Gyarmati 2

THEOREMA EGREGIUM preliminary Solve these equations: I. II. ; ;         THE RESULT:         Corollary: on the surface exist an internal, uncontradictory geometry, which 1. Euclidean only if the surface is plain; 2. non-Euclidean, if not. distance dr. prof. P. G. Gyarmati 3

THEOREMA EGREGIUM Gauss: Disquisitiones generales circa superficies curves. 1827. Commentat. Soc. Göttingensis, VI. I. Surface with tangent in a point and the gradient II. Place a plain along the gradient III. The plain cuts a plain-curve from the surface IV. Rotate the plain around the gradient as an axis V. Each move cuts different plain-curves VI. Each plain-curve have different radius at the point VII. After full turn get a min. R1 and a max. R2       Define the curvature as and generally Results: k=0  plain surface; k=+const. regular surface, convex; k=-const.  regular surface, saddle k = 1, then the geometry spherical: k = 0, then the geometry Euclidean: k =-1, then the geometry hyperbolic: curvature dr. prof. P. G. Gyarmati 4

THEOREMA EGREGIUM corollary From the area of a triangle: k=1 ; k=0 ; k=-1 Comments for the future: - Riemann space: ; Riemann tenzor: - At Gauss (2 dimensions): and - n=4  Minkowski space - Einstein gravitational equotations Results and future dr. prof. P. G. Gyarmati 5

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