Our Euclidean intuition, probably inherited from ancient primates, might have grown out of the first seeds of geometry in the motor control systems of early animals who were brought up to sea and then to land by the Cambrian explosion half a billion years ago. The primates' brain had been idling for 30-40 million years. Suddenly, in a flash of one million years, it exploded into growth under the relentless pressure of sexual-social competition and sprouted a massive neocortex (70% neurons in humans) with an inexplicable capability for language, sequential reasoning and generation of mathematical ideas. Then Man came and laid down space on papyrus in a string of axioms, lemmas and theorems around 300 B.C. in Alexandria.
Projected to words, the brain's model of space began to evolve by dropping, modifying and generalizing its axioms. The Parallel Postulate fell first: Gauss, Schweikart, Lobachevski,1 Bolyai (who else?) came to the conclusion that there is a unique non-trivial one-parameter deformation of the metric on \(\R^3\) keeping the space fully homogeneous.
It is believed, that Gauss, who convinced himself of the validity of
hyperbolic geometry somewhere between 1808 and 1818, was disconcerted by
the absence of a Euclidean realization of the hyperbolic plane \(H^2\). By that
time, he must have had a clear picture of the geometry of surfaces in \(\R^3\) (exposed in his
Disquisitones circa superficies curvas
in \(1827\)), where the
(intrinsic) distance between two points on a surface is defined as the length of the shortest (better to say innmal
) curve in the surface between these
points. (This idea must have been imprinted by Nature in the brain, as
most animals routinely choose shortest cuts on rugged terrains.) Gauss discovered the following powerful and efficient criterion for isometry between
surfaces, distinguishing, for example, a piece of a round sphere \(S^2\subset \R^3\) from an arbitrarily bent sheet of paper (retaining its intrinsic Euclideanness
under bendings).
Map a surface \(S\subset \R^3\) to the unit sphere \(S^2\) by taking the vectors \(\nu (s) \in S\) parallel to the unit normal vector vectors \(v(s)\), \(s\in S\). If \(S\) is \(C^2\)-smooth, the Gauss map \(G: S\to S^2\,, s\mapsto \nu(s)\) is \(C^1\) and its Jacobian, i.e. the infinitesimal area distorsion, comes with a non-ambiguous sign [...] and so \(S\) appears with a real function, called Gauss curvature \(\Kappa(s) = \operatorname{det}(J_G(s)) \).
Every isometry between surfaces, say \(f: S \to S'\), preserves Gauss curvature, \(\Kappa(f(s)) = \Kappa(s)\) for all \(s\in S\).
For example, the plane has \( \Kappa \equiv 0\)(as the Gauss map is constant) and so it is not (even locally) isometric to the unit sphere where \(\Kappa \equiv 1\)(for the Gauss map is identity on \(S^2\)). More generally, no strict convex surface is locally isometric to a saddle surface, such as the graph of the function \(z = xy\) for instance, since trict convexity makes \(\Kappa>0\) while saddle points have \(\Kappa \leqslant 0 \).
Gauss was well aware of the fact that the hyperbolic plane \(H^2\) would have constant negative curvature if it were realized by a surface in \( \mathbb{R}^3 \). But he could not find such a surface! In fact, there are(relatively) small pieces of surfaces with \(\Kappa = -1\) in \(\mathbb{R}^3\) investigated by Beltrami in \(1868\) and it is hard to believe Gauss missed them;but he definitely could not realize the whole \( H^2 \) by a \(C^2-\)surface in \(\mathbb{R}^3\) (as is precluded by a theorem of Hilbert(\(1901\)).) This could be why(besides his timidity in the face of the Kantian guards of Trilobite's intuition) Gauss refrained from publishing his discovery.
Probably, Gauss would have been dlighted to learn(maybe he knew it?) that the flat Lorenz-Minkowski metric
\(dx^2 + dy^2 - dz^2\) on \(\mathbb{R}^{2, 1} = \mathbb{R}^3\) induces a true
positive metric on the sphere
where each of the two components of \( S^2_{-} \)(one is where \(z >0\) and the other with \(z<0 \)) is isometric to \(H^2\) and where the orthogonal group \(O(2, 1)\)(i.e. the linear group preserving the quadratic form \(x^2 + y^2 - z^2\) ) acts on these two \(H^2\)'s by (hyperbolic) isometrics.
There is no comparable embedding of \(H^2\) into any \(\mathbb{R}^N\)(though \(H^2\) admits a rather contorted isometric \(C^\infty\)-immersion to \(\mathbb{R}^5\)(to \( \mathbb{R}^4\)?) and, incredibly, an isometric \(C^1-\)embedding into \(\mathbb{R}^3\)) but it admits an embedding into the Hilbert space, say \(f:H^2 \to \mathbb{R}^{\infty}\), where the induced intrinsic metric is the hyperbolic one, where all isometries of \(H^2\) uniquely extend to those of \(\mathbb{R}^\infty\) and such that $$ \operatorname{dist}_{\mathbb{R}^\infty}(f(x), f(y)) = \sqrt{\operatorname{dist}_{H^2}(x, y)} + \delta(\operatorname{dist}(x, y)) $$ with bounded function \(\delta(d)\)(where one can find \(f\) with \(\delta(d) = 0\), but this will be not isometric in our sense as it blows the lenghts of all curves in \(H^2\) to infinity.) Similar embeddings exist for metric trees as well as for real and complex hyperbolic spaces of all dimensions but not for the other irreducible symmetric spaces of non-compact type.(This is easy for trees: arrange a given tree in \(\mathbb{R}^3\), such that its edges become all mutually orthogonal and have prescribed lengths).
Summary. Surfaces in \(\mathbb{R}^3\) provide us with a large easily accessible pool of metric spaces: take a domain in \(\mathbb{R}^2\), smoothly map it into \(\mathbb{R}^3\) and, voilà, you have the induced Riemannian metric in your lap. Then study the isometry problem for surfaces by looking at metric invariants(curvature in the above discussion), relate them to standard spaces (\(\mathbb{R}^N, \mathbb{R}^\infty, \mathbb{R}^{2, 1}\)), and consider interesting(to whom?) classes of surfaces, e.g. those with \(\Kappa>0\) and with \(\Kappa <0 \).