1. 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

2. 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

3. 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

4. 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

5. THEOREMA EGREGIUM corollary
From the area of a triangle:
k=1 ; k= ; 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

6. THANKS FOR YOUR ATTENTION!
© 2017 Dr. Prof. Peter G. Gyarmati
BGL-2017, Gyöngyös