# Meaning of a cryptic sentence by Gauss on “the mobility of figures in the hyperbolic plane”

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.

• 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. – Conifold Mar 28 at 19:36
• 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/… ? – user2554 Mar 28 at 20:12
• 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? – Conifold Mar 28 at 20:21

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 $$v_2$$.