Is there an absolute geometry that underlies spherical, Euclidean and hyperbolic geometry?

Question

A space form is defined as a complete Riemannian manifold with constant sectional curvature. Fixing the curvature to +1, 0 & -1 and then taking the universal cover by the Killing–Hopf theorem yields spherical, Euclidean and hyperbolic space. This is a uniform analytic construction of these three fundamental spaces.

Now all three of these spaces have a synthetic description: Euclidean geometry by Euclid, Hyperbolic geometry by Lobachevsky and spherical geometry by Riemann.

Q. Is there a uniform description of all three in that there is an 'absolute geometry' that underlies them?

To explain what I mean by this, recall it was Bolyai who introduced the term 'absolute geometry' by dropping the parallel axiom from Euclid's plane geometry called the remaining axiomatic system this name. It turns out we can still construct parallel lines here and hence it cannot underlie spherical geometry. However, it underlies both the Euclidean and hyperbolic plane. According to Wikipedia's page on this topic, it is possible to modify the axioms so we also include spherical geometry as well as "elliptic geometry" which I think is the geometry of the real projective plane. They don't give any details but do refer to Ewald 1971, Geometry: An Introduction.

I don't have access to this book, can someone familiar with it please verify Wikipedia's claim and perhaps give a summary of the axioms used if so.

Answer 1

I don't know if this is the kind of answer you expect, but:

In the hyperbolic space of dimension $n+1$ one naturally gets all $n$-dimensional constant curvature geometries.

• spheres (points at distance $\le r$ from a given point) inherit their spherical $n$-dimensional geometry. I'm not sure what the curvature is as a function of $r$, but it tends to $0$ resp. $\infty$ when the radius tends to $\infty$ resp. $0$.
• horospheres inherit the Euclidean $n$-dimensional metric. Recall that a horofunction is a limit $h$ of functions $x\mapsto d(x,x_n)-d(x_0,x_n)$ for some sequence $x_n$ tending to infinity, and a horosphere is $\{x:h(x)=0\}$ for some horofunction $h$.
• for two distinct points $x_1,x_2$, the set of $x$ such that $d(x,x_1)=d(x,x_2)$ inherits the $n$-dimensional hyperbolic geometry.

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Edit: One can even fully interpolate: consider the half-space model $H$ in $\mathbf{R}^{n+1}$. For a round sphere or hyperplane $S$ in $\mathbf{R}^{n+1}$ with $S\cap H\neq\emptyset$, the intersection $S\cap H$ achieves all these cases: either $S\subset H$, hence is a sphere in $H$ as well (not with the same center!), with all possible positive curvature. If $S$ is tangent to $\partial H$ (or $S$ is parallel to $H$), then it is a horosphere, i.e. with zero curvature. If $S$ is not contained in $\bar{H}$, then one gets something with constant negative curvature (which is totally geodesic when $S$ meets $\partial H$ in a perpendicular way). It seems however that the negative curvature cannot be arbitrarily close to $-\infty$, by a compactness argument. Possibly it is $\ge -1$ with equality exactly in the geodesic case?

Answer 2

The question is what to modify and how much one is willing to modify. As everybody who seriously thought about elementary geometry knows, Euclid's axioms are inadequate. There are several "standard" alternative axiomatics, most common is Hilbert's. Marvin Greenberg in Appendix A to Chapter 10 of his book

Greenberg, Marvin Jay, Euclidean and non-Euclidean geometries. Development and history, New York, NY: W. H. Freeman and Company (ISBN 978-0-7167-9948-1). xxix, 637 p. (2008). ZBL1127.51001.

modifies Hilbert's primitive notion of betweenness in Euclidean geometry (he replaces it with a primitive notion of separation). - Greenberg explains why this modification is needed if one were to get the elliptic geometry. With this modification, the geometry of the projective plane (aka the "elliptic geometry") is then obtained via a "negation" of the Playfair's axiom (the negation is that "no parallels exist"). You can find details by reading Greenberg's book which you should be able to find in a nearby library. I am unaware of a similar modification of Birkhoff axioms and Tarski axioms (and I do not see how it can be done without a complete revision of their axiomatics).

You can then regard Greenberg's modified form of Hilbert's axioms, minus the Playfair's axiom, as a list of axioms of an absolute geometry that "underlies" hyperbolic, euclidean and elliptic geometry. One can, in principle, also get the spherical geometry, but that requires dropping an additional axiom. Maybe this is what you are interested in.

Answer 3

I'm not sure exactly what you mean by 'absolute geometry' that unifies spherical, Euclidean, and hyperbolic geometry, but in the sense of Klein's identification of geometries with group actions, so that an 'underlying' geometry is an extension of a group action, there is a very specific answer to your question.

There are two distinct geometries that 'underlie' each of the spherical, Euclidean, and hyperbolic geometries in dimension $n$: conformal geometry and projective geometry. Moreover, there is no finite dimensional Lie transformation group that extends either of these geometries.

In the case of conformal geometry, the simply-connected model space is $\mathbb{S}^n$, regarded as the quotient of $\mathrm{O}^+(n{+}1,1)$ by the subgroup that fixes a future-directed null ray in the Lorentzian space $\mathbb{R}^{n{+}1,1}$, i.e., $\mathbb{S}^n$ is the space of future-directed null rays in $\mathbb{R}^{n{+}1,1}$. To specialize to the spherical case, one takes the subgroup $G_+\subset\mathrm{O}^+(n{+}1,1)$ that fixes a future-directed timelike vector $v_+\in\mathbb{R}^{n{+}1,1}$. Then $G_+$ acts transitively on $\mathbb{S}^n$ and preserves a metric of positive sectional curvature $+1$ on $\mathbb{S}^n$. To specialize to the Euclidean case, one takes the subgroup $G_0\subset\mathrm{O}^+(n{+}1,1)$ that fixes a future-directed null vector $v_0\in\mathbb{R}^{n{+}1,1}$ and notes that $G_0$ acts transitively on the set $\mathbb{S}^n\setminus \{\mathbb{R}^+v_0\}$, preserving a (flat) Euclidean metric. To specialize the hyperbolic case, one takes the subgroup $G_-\subset\mathrm{O}^+(n{+}1,1)$ that fixes a spacelike vector $v_-\in\mathbb{R}^{n{+}1,1}$ and notes that $G_-$ acts transitively on the hemisphere $\mathbb{H}^n = \{ \mathbb{R}^+v\in \mathbb{S}^n | \langle v_-,v\rangle <0 \} $ preserving a Riemannian metric of sectional curvature $-1$.

In the case of projective geometry, the simply-connected model space is $\widetilde{\mathbb{RP}^n}$, the space of rays in $\mathbb{R}^{n+1}$, regarded in a natural way as the quotient of $\mathrm{PGL}(n{+}1,\mathbb{R})$ by the subgroup that fixes a specific ray $r_0\in \mathbb{R}^{n+1}$. Spherical geometry is recovered by restricting to the subgroup $\mathrm{PO}(n{+}1)\subset \mathrm{PGL}(n{+}1,\mathbb{R})$, which acts transitively on $\widetilde{\mathbb{RP}^n}$ preserving a Riemannian metric with sectional curvature $+1$, and hyperbolic geometry is recovered by restricting to the subgroup $\mathrm{PO}(n,1)\subset \mathrm{PGL}(n{+}1,\mathbb{R})$, which acts transitively on a hemisphere of $\widetilde{\mathbb{RP}^n}$ preserving a Riemannian metric with sectional curvature $-1$, etc.

Note that, while $\mathbb{S}^n$ and $\widetilde{\mathbb{RP}^n}$ are diffeomorphic, the two groups $\mathrm{O}^+(n{+}1,1)$ and $\mathrm{PGL}(n{+}1,\mathbb{R})$ are not isomorphic when $n>1$. They don't even have the same dimension. Hence, these two 'underlying geometries' are not equivalent in any sense. Moreover, it is not hard to show that the only Lie transformation group on $\mathbb{S}^n$ that properly contains $\mathrm{O}^+(n{+}1,1)$ is the full group of diffeomorphisms of $\mathbb{S}^n$ and that the only Lie transformation group on $\widetilde{\mathbb{RP}^n}$ that properly contains $\mathrm{PGL}(n{+}1,\mathbb{R})$ is the full group of diffeomorphisms of $\widetilde{\mathbb{RP}^n}$. Thus, these two incompatible underlying geometries are, in a sense, 'minimal' nontrivial underlying geometries.