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G. Waldo Dunnington writes in pages 189-190 of his biography of Gauss:

Among the axioms of geometry which do not depend on the parallel postulate are those which secure the free mobility of a figure in space. This means that space is the same in nature throuoght and that any figure can be carried along it without tearing or crumpling or stretching ... ... In a note dated about 1827 Gauss called the curved surface of constant negative measure of curvature generated by rotation of the tractrix (pseudo sphere) the "counterpart of the sphere". The formulas set up by him lead to the theorem that in the pseudo sphere (and only in it) surfaces of rotation are congruent to each other and that, preserving this property, one can move a geodetic triangle on the pseudo sphere just as one can move a spherical triangle on a sphere.

I have also found mentions of such sentences in Paul Stackel's treatise on Gauss's contributions to geometry.

I have three questions on this quoted passage:

  • First of all, what axioms does Dunnington refers to when he says "those which secure the free mobility of figures in space"? are there really "free mobility axioms"?
  • What's the meaning of "in the pseudosphere surfaces of rotation are congruent"? as far as i know the pseudosphere is a surface (not a space!) so how can one talk about surfaces contained in it? this sentence seems completely unrelated and obscure.
  • What's the meaning of "moving a geodetic triangle on the pseudosphere"? In euclidean geometry the rigidity of motion is taken as granted so one can speak about motion of figures without distortion, but how can one speak the same about hyperbolic geometry? i really want to understand this point.

Every usefull comment will be blessed.

Update:

I think the article "What did Gauss read in the Appendix?" might be very useful in answering this question. In pages 22-24 of this article the authors analyze the relevant (to our question) fragment from Gauss's nachlass (vol 8, p.255-257; Stackel refers to these pages when he comments on Gauss's formulas about the "mobility of figures") - in this fragment Gauss assumed a hyperbolic triangle which changes with time (so one can view it as a kind of "motion") and analyzed the metric and angular relations of it by the process of functional equations.

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    $\begingroup$ Free mobility is the term used by Helmholtz and Riemann, it roughly means homogeneity and isotropy, Helmholtz axiomatized it. One can literally move geodesic triangles congruently along the pseudosphere because it has constant curvature. "Surfaces of rotation are congruent" should read "figures under rotation are congruent", probably a typo. $\endgroup$ – Conifold Mar 28 at 19:36
  • $\begingroup$ I tend to agree with you about the second question (i just wanted to make sure i'm not missing something). But for my third question i'm looking for more mathematically oriented answer - isn't it related to this question mathoverflow.net/questions/303043/… ? $\endgroup$ – user2554 Mar 28 at 20:12
  • $\begingroup$ One can derive constant curvature from Helmholtz's axioms, but this is not the right site for that. You can check out references under What theorem of Sophus Lie on the number of geometries is H. Poincaré referring to? $\endgroup$ – Conifold Mar 28 at 20:21
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The answers to your first and third questions are the following. "Free mobility" of a space means that there is a (pseudo-) group of (local) isometries (1-1 maps preserving distances) which acts properly and transitively on pairs (point,orthonormal basis in the tangent space). For surfaces, this means that whenever you have $(p_1,v_1)$ and $(p_2,v_2)$ where $p_j$ are points of the space and $v_j$ are directions (unit tangent vectors) at these points, there is a unique isometry of the space which sends $p_1$ to $p_2$ and $v_1$ to $v_2$.

In particular on each surface of constant curvature there is such a group. If you have a triangle on a surface, you can move it to other place so that angles and lengths of sides are preserved. In Euclid this axiom is not stated explicitly but it is used all the time in the form that "triangles with two sides and angle between the equal are equal." The word "equal" in Euclid means "can be moved to coincide", that is there is a unique element of the group mentioned above which sends one object to another. It is fundamental for geometry.

The problem of classifying all possible (local) geometries was stated by Riemann and Helmholtz and solved by Killing. It is called the Riemann-Helmholtz problem. In particular, there are three types of geometries in dimension 2 (spherical, hyperbolic and usual, Euclidean).

To your 2-nd question I cannot answer, it is indeed unclear what the author says.

General remark. If you want to learn mathematics, I do not recommend the books written by historians. Read books written by mathematicians. Historians use imprecise language, and frequently do not really understand themselves what they are writing about. Also a Gauss biography is usually a poor source for his mathematics, though it can be a good source about his life. Most biography writers do not really understand mathematics, and rely other sources to describe it, with inevitable distortions.

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