Felix Klein · Book 7: Scientist Gallery

Part I · 1859–1871 · Distance Out of a Conic

The algebraic road to the same theorem

Beltrami built a piece of the hyperbolic plane out of a surface. Klein built the whole of it out of a quadratic form, and the two constructions are the same theorem reached from opposite directions — one differential, one linear.

The machinery is Cayley’s. In the Sixth Memoir upon Quantics of 1859 he fixed a conic in the projective plane, called it the absolute, and defined the distance between two points as a constant multiple of the logarithm of the cross-ratio they form with the two points where their line meets that conic. Angles the same way, dually. The whole metric is a function of one quadratic form and the projective structure, and Cayley drew the conclusion in a sentence that is still the shortest statement of the programme: metrical geometry is part of descriptive geometry, and descriptive geometry is all geometry.

One matrix, three geometries

Klein saw in 1871 what the choice of absolute controls. Take the conic $x^{2}+y^{2}-z^{2}=0$ and the interior points become a model of the hyperbolic plane, with chords of the disc as its straight lines — complete, unlike the pseudosphere, at the cost of no longer being a surface you can hold. Take an imaginary absolute and the same construction returns elliptic geometry. Degenerate the conic and Euclidean geometry falls out as the limiting case. The signature of a symmetric bilinear form selects which of the three you are doing, and nothing else in the construction changes. That is as close as geometry comes to a classification by linear algebra alone.

The isometries come along for free. In the upper half-plane picture the hyperbolic isometries are the action of $\mathrm{PSL}(2,\mathbb{R})$ by Möbius transformations — two-by-two real matrices of determinant one, modulo sign. A geometry that took two thousand years to admit exists is generated by a group you can write on a postcard.

Part II · 1872 · Erlangen

Geometry as the invariant theory of a group

The inaugural programme at Erlangen turned the observation into a definition. Given a space and a group acting on it, the geometry is the theory of the properties left invariant by that action. Euclidean geometry is the invariant theory of the rigid motions; affine geometry of the affine group; projective geometry of the projective group; hyperbolic geometry of the subgroup preserving the absolute. Different geometries are different subgroups of one group, and the inclusions between the subgroups are the relations between the geometries.

This is the move the corpus makes constantly and should name once: stop describing the objects and describe what is allowed to move them. A property is geometric when it survives the group. The classification problem becomes a lattice of subgroups.

Where Erlangen stops, and it stops early

The programme covers the homogeneous geometries — the ones with enough symmetry that the group acts transitively. Generic Riemannian manifolds have no symmetry group at all: a surface with a bump has isometry group the identity, and Erlangen has nothing to say about it. That class is exactly where Beltrami’s other contribution lives, since $\Delta_{g}$ is defined on any metric whatever and needs no group. Cartan’s later work on connections exists to bridge the gap, and the honest summary is that Erlangen is a magnificent programme for the symmetric case and silent outside it. A corpus that reaches for group invariants should know which of its objects are homogeneous before the reflex fires.

Beltrami and Klein prove the same thing about the parallel postulate and disagree about what geometry is made of. One says: a metric, and whatever curvature it happens to have. The other says: a group, and whatever it happens to fix. Neither answer contains the other, and both are still in use — the first became Riemannian geometry and general relativity, the second became representation theory and gauge theory.

— the 1868 and 1872 programmes, set side by side

Place in the Series

The linear algebra here is not decoration and it is not advanced: a symmetric bilinear form, its signature, and a group preserving it. That is the whole apparatus, and the corpus leans on the same three notions wherever a threshold, a ratio or an invariant appears. A quadratic form appears again in ch-beltrami as the first fundamental form, and again under the hex grid, where the axial coordinates carry $Q(q,r)=q^{2}+qr+r^{2}$ — positive definite, discriminant $-3$, with the six neighbours as its minimal vectors. Three appearances of one object, none of them yet stated in a chapter of their own.

References

Cayley 1859	A Sixth Memoir upon Quantics. Phil. Trans. R. Soc. 149, 61–90 — the absolute, and distance from cross-ratio.
Klein 1871	Ueber die sogenannte Nicht-Euklidische Geometrie. Math. Ann. 4, 573–625.
Klein 1872	Vergleichende Betrachtungen über neuere geometrische Forschungen — the Erlangen programme.
Cartan 1920s	Connections and moving frames — the extension Erlangen needs to reach non-homogeneous spaces.
Verification	`book7/ch-felix-klein-verify.py` — 4 blocks. Measures the angle distortion of Klein’s metric point by point and shows Poincaré’s has none — which is why Circle Limit is the Poincaré disk. Ends with an `[HONESTY]` block naming what is evidence rather than proof.
Internal	ch-beltrami (the differential route to the same theorem) · ch-escher (the disc model, drawn)