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Geometry: Geometry is the branch of mathematics that deals with the shapes, sizes, and positions of figures. See also Mathematics, Arithmetics, Forms.
Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

Author Concept Summary/Quotes Sources

Bernulf Kanitscheider on Geometry - Dictionary of Arguments

I 187f
Geometry/cosmology/FRW/Kanitscheider: a peculiarity of the spatial part of the line element is in the kind of the time-dependent geometry which it includes: the factor R(t),provides for the fact that all spatial structures of cosmic extent (thus larger than galaxies) experience a rotation or shrinking. (Whereby a triangle remains similar).
The homogeneity condition excludes other geometrical changes (e.g. shearing of a triangle, whereby the area, but not the form would remain). ((s) Thus the density would increase or decrease).
Furthermore, the isotropy forbids a rotation, whereby a direction would be distinguished.
Kanitscheider: however, all this goes back to the underlying boundary conditions is not logically a priori or physically necessary!
Also world models with relaxed boundary conditions and thus shearing and rotation are to be brought in agreement with Einstein's field equations.
>Field equations
I 188/189
Curvature/channel separator: free parameter: k, the so-called curvature index.
k: curvature index.
At k = 0 the physical space is flat, Euclidean. Parabolic.
At k = +1 it is spherical. Compact, closed world. Trisphere, a most distant point. Unique and invertible mapping, connected to the trisphere S³. Again, an elliptic relation can be obtained by identifying the antipodal points, although the volume of the spaces may well be different.
For k = -1 hyperbolic, topologically ambiguous: Euclidean dimensional relations apply locally on both the cylinder and the cone, i.e. finite and infinite models are possible. Also here (and at k = 0) one can achieve by an identification of antipodal points that the three-space becomes compact, only here thereby the symmetry properties of the space are roughly changed, they are then no longer isotropic.
But this actually does not answer the question whether the space is infinite, but the line element always only determines the local metric geometry!
However, it can be said independently that the world is undoubtedly infinite, i.e. it has no spatial boundary.
>Universe/Kanitscheider, >Space curvature/Kanitscheider, >Relativity theory.

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments
The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.

Kanitsch I
B. Kanitscheider
Kosmologie Stuttgart 1991

Kanitsch II
B. Kanitscheider
Im Innern der Natur Darmstadt 1996

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