List of Erlangen Program Geometries

Introduction

This article lists the geometries of the corpus in the form in which Klein's principle presents them: a geometry is the study of the invariants of a transformation group, and a geometric object is a figure on which the group acts. The list gathers the transformation-group form of Euclidean, affine, projective, hyperbolic, conformal, Möbius and symplectic geometry, together with the further homogeneous geometries of the corpus, the correspondence between each geometry and its automorphism group, and the invariant that the group of each geometry preserves. Every entry points to the article that introduces the geometry or the group, and the article introduces nothing and proves nothing.

The organising relation is the correspondence between a geometry and its automorphism group: the group acts transitively on the space, the stabiliser of a point is the subgroup that fixes it, and the space is recovered as the coset space $G/H$. The Erlangen principle belongs to the planned article Transformation Groups and the Erlangen Program; its statement is named here and pointed to, and the general theory of transformation groups is that of Transformation Groups.

The article records examples and non-examples side by side. Beside the geometries that a transformation group describes it lists the geometric theories whose isometry group is too small to describe them, the structure groups of bundles that are not transformation groups of a space, and the groups that are defined by a form rather than by an action, each with the failure named and the article that records it.

Klein's Principle and Transformation Groups

The principle states that a geometry is the study of the invariants of a transformation group, so that the group is the primary datum and the geometric notions are those the group preserves.

Object	The property it has	Introduced in
Klein's principle (the Erlangen program)	a geometry is the study of the invariants of a transformation group	Transformation Groups and the Erlangen Program (planned)
a transformation group	a group acting faithfully on a set with structure, through a permutation representation	Transformation Groups
the action $G\times X \to X$	the map organising the geometry, with orbits and stabilisers	Transformation Groups
the automorphism group $\operatorname{Aut}(X)$	the group of automorphisms of a structure on $X$	Transformation Groups
the orbit $\operatorname{Orb}(x)$ and stabiliser $\operatorname{Stab}(x)$	the orbit and the point stabiliser, related by the orbit–stabiliser theorem	Transformation Groups
the coset space $G/H$	a transitive $G$-set, the homogeneous form of a geometry	Transformation Groups; Homogeneous Spaces
a measure-preserving transformation group	a group of automorphisms of a measure space, whose invariant is the measure	Ergodic Theory

The ideal of the principle is that every geometric notion is a notion invariant under the group and conversely that every invariant of the group has a geometric meaning. The two classical cases at the extremes are the symmetric group $\operatorname{Sym}(X)$, under which no nontrivial structure is invariant, and the full group of all homeomorphisms of a manifold, under which only the topological invariants are invariant; the geometries of the list are the intermediate cases in which a form, a distance or a conformal or projective structure is preserved.

The Correspondence between a Geometry and its Automorphism Group

Each geometry of the list carries a group, and the geometry is recovered from the group together with the subgroup fixing a point.

Geometry	The transformation group it is the invariant theory of	Introduced in
a homogeneous geometry	the pair $(G,H)$ of a group and a stabiliser, with the space $G/H$	Homogeneous Spaces
the transitivity of a geometry	the group acts transitively, so the space is a single orbit	Transformation Groups; Homogeneous Spaces
the invariant of a geometry	a function or tensor on the space fixed by the group action	Transformation Groups
the group of a geometry	the automorphism group of the structure, recovered as $\operatorname{Aut}(X, \text{structure})$	Transformation Groups
the linear model of a geometry	the isotropy representation at a point, a linear representation of $H$	Homogeneous Spaces

The correspondence is exact when the geometry is homogeneous and the automorphism group is the full structure-preserving group: the space is the coset space, the stabiliser is the isotropy subgroup, and the local geometry is the isotropy representation. The hierarchy $\operatorname{Sym}(X) \supset \operatorname{Aut}(X, \text{structure}) \supset \cdots$ of the transformation-group article is the ordering of the geometries from the weakest to the strongest structure.

The Classical Geometries as Instances

The classical geometries are the standard instances of the principle, each with the group whose invariants it studies.

Geometry	Its transformation group	Introduced in
Euclidean geometry	the Euclidean group $E(n) = \mathbb{R}^n \rtimes O(n)$	Euclidean Geometry
affine geometry	the affine group $\operatorname{Aff}(n) = GL(n) \ltimes \mathbb{R}^n$	Affine Spaces and Translations
projective geometry	the projective linear group $PGL(n+1,\mathbb{K})$	Projective Geometry
hyperbolic geometry	the isometry group $O(n,1)$, orientation-preserving part $PSL(2,\mathbb{R})$ in dimension two	Hyperbolic Geometry
spherical geometry	the orthogonal group $O(n+1)$	Spherical Geometry
conformal geometry	the conformal group $O(n+1,1)/\{\pm 1\}$	Conformal Geometry
Möbius geometry	the Möbius group $\operatorname{Möb}(n) = O(n+1,1)/\{\pm 1\}$	Möbius and Lie Sphere Geometry
symplectic geometry	the symplectic group $Sp(2n,\mathbb{R})$, preserving the form $\omega$	Symplectic Geometry; The Unitary and Symplectic Groups
the linear classical geometry	the general linear group $GL(n,\mathbb{K})$ and its special, orthogonal, unitary and symplectic subgroups	Matrix Groups and Classical Groups
the geometry of a quadratic form	the orthogonal group $O(V,Q)$, the invariants of the form	Isometries and Orthogonal Transformations

The instances are ordered by the size of the group: the general linear group gives the largest geometry and fewest invariants, the Euclidean and hyperbolic groups the metric geometries with distance and angle, the conformal group the angle alone, and the symplectic group a single alternating form. Each group is the automorphism group of the corresponding structure, so each geometry is the invariant theory of that group, which is the content of the principle.

The Invariants of the Classical Geometries

The invariant that a geometry studies is the datum its group preserves, and the list of groups with their invariants is the concrete form of the Erlangen correspondence.

Geometry	The invariant it studies	Introduced in
projective geometry	the cross ratio of four collinear points	Projective Geometry, §Coordinates, Frames and the Cross Ratio
Euclidean geometry	distance, angle, area and the congruence of figures	Euclidean Geometry
affine geometry	ratios of lengths on a line and the incidence of parallel lines	Affine Spaces and Translations
spherical geometry	the spherical distance and the spherical angle	Spherical Geometry
hyperbolic geometry	the hyperbolic distance and the hyperbolic angle	Hyperbolic Geometry
conformal geometry	the angle, preserved although the length is not	Conformal Geometry
Möbius geometry	the cross ratio of four points and the generalised spheres	Möbius and Lie Sphere Geometry
symplectic geometry	the symplectic form $\omega$ and the Hamiltonian vector field it defines	Symplectic Geometry

The invariant theory of a geometry is not the whole of the geometry of that space: the differential geometry of a manifold uses the metric and its derivatives, which are not invariants of a finite-dimensional group, and the topological invariants are those of the homeomorphism group. The distinguishing feature of an Erlangen geometry is that its group is a finite-dimensional Lie group, so that its local invariants are finite in number and computable from the isotropy representation.

Homogeneous Geometries Beyond the Classical List

The correspondence extends beyond the classical geometries to every space whose group acts transitively, with the stabiliser of a point supplying the local model.

Geometry	Its transformation group	Introduced in
the geometry of a symmetric space	the isometry group $G$, with isotropy the fixed-point group $K$ of the geodesic symmetry	Symmetric Spaces
the geometry of a flag manifold	the general linear group acting transitively on the flags, with the parabolic stabilisers	Flag Manifolds
the geometry of a Grassmannian	the orthogonal or the general linear group acting transitively on the subspaces	Grassmannians and Stiefel Manifolds
the geometry of a reflection group	the group generated by the reflections in the walls of a chamber	Coxeter Groups

Each of these is a homogeneous geometry in the exact sense of the principle: the space is the coset space of the group by the stabiliser, the invariant theory is the invariant theory of the group, and the local model is the isotropy representation. The symmetric spaces and the flag manifolds are the families in which the correspondence is developed in the corpus, and the reflection groups are the discrete counterpart, whose chambers are the fundamental domains of the group action.

Warnings

An object that a reader may expect among the Erlangen program geometries, and does not find, is recorded with the reason.

Object	Why it is not listed as an Erlangen geometry	Introduced in
the Riemannian geometry of a general manifold	its isometry group need not act transitively, so no transformation group describes it	Riemannian Geometry; Homogeneous Spaces
spin geometry and the group $\operatorname{Spin}(n)$	a structure group of a bundle, not the transformation group of a space	Spin Geometry; The Clifford, Pin and Spin Groups with Signed Inner Conjugation
the orthogonal group $O(n)$ as a geometry	a group defined by a form; the geometry it belongs to is the orthogonal geometry of the form	Isometries and Orthogonal Transformations
the symplectomorphism group	the automorphisms of a symplectic manifold, generally larger than the finite-dimensional $Sp(2n,\mathbb{R})$	Diffeomorphism Groups; Symplectic Geometry
the conformal group in dimension two	infinite-dimensional, so the invariant theory is not the finite-dimensional one	Complex Analysis; Conformal Geometry

Summary

This article has listed the geometries of the corpus in the form of Klein's principle. The principle and the transformation group open the list, with the action, the orbit, the stabiliser and the automorphism group; the correspondence between a geometry and its automorphism group follows, with the homogeneous space $G/H$ and the isotropy representation as the local model; the classical geometries are the instances, Euclidean, affine, projective, spherical, hyperbolic, conformal, Möbius and symplectic, each with the group whose invariants it studies; and the invariants themselves — the cross ratio, the distance and angle, the conformal angle, the symplectic form and the invariant measure — close it, and the homogeneous geometries beyond the classical list — the symmetric spaces, the flag manifolds, the Grassmannians and the reflection groups — show the correspondence at work outside it. Beside the examples stand the non-examples: a general Riemannian manifold has no transitive transformation group, spin geometry is a structure group rather than a transformation group, and the symplectomorphism group is larger than the finite-dimensional symplectic group.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following, and they are the symbols of the articles that introduce them.

Symbol	Meaning
$\operatorname{Sym}(X)$, $\operatorname{Aut}(X)$	Symmetric group, automorphism group of a structure
$\operatorname{Orb}(x)$, $\operatorname{Stab}(x)$, $G/H$	Orbit, stabiliser, coset space of a transitive action
$E(n)$	The Euclidean isometry group, $\mathbb{R}^n\rtimes O(n)$
$GL(n,\mathbb{K})$, $PGL(n+1,\mathbb{K})$	General linear and projective linear groups
$O(n+1)$, $O(n,1)$, $O(n+1,1)$	Spherical, hyperbolic and conformal groups
$\operatorname{Möb}(n) = O(n+1,1)/\{\pm 1\}$	Möbius group
$Sp(2n,\mathbb{R})$, $\omega$	Symplectic group and symplectic form
$O(V,Q)$	Orthogonal group of a quadratic form
$\operatorname{Aff}(n)$	The affine group, $GL(n)\ltimes\mathbb{R}^n$
$\operatorname{Spin}(n)$	The spin group, the double cover of $SO(n)$