(G, X) structure

In mathematics , (G, X) structures (also locally homogeneous structures or geometric structures ) provide the possibility of providing topological manifolds with geometric structures in the sense of Felix Klein's Erlangen program . This approach is used in the geometrization of 3-manifolds and in the representation theory of groups .

(G, X) structures

Let it be a Lie group and a transitive G-space . ${\displaystyle G}$ ${\displaystyle X}$

A -manifold is a manifold with an -atlas , i.e. a cover by open sets ${\displaystyle \left(G,X\right)}$ ${\displaystyle M}$ ${\displaystyle \left(G,X\right)}$ ${\displaystyle \left\{\left(U_{i},\phi _{i}\right):i\in I\right\}}$

${\displaystyle M=\bigcup _{i\in I}U_{i}}$

along with homeomorphisms

${\displaystyle \phi _{i}\colon U_{i}\rightarrow \phi _{i}\left(U_{i}\right)\subset X}$

on open subsets of such that all coordinate transitions ${\displaystyle X}$

${\displaystyle \gamma _{ij}=\phi _{i}\phi _{j}^{-1}\colon \phi _{j}\left(U_{i}\cap U_{j}\right)\rightarrow \phi _{i}\left(U_{i}\cap U_{j}\right)}$

Constraints from elements are. ${\displaystyle G}$

Development mapping and holonomy

Development illustration

Fix a base point and a map with . Be ${\displaystyle x_{0}\in M}$ ${\displaystyle \left(U_{0},\phi _{0}\right)}$ ${\displaystyle x_{0}\in U_{0}}$

${\displaystyle \pi :{\widetilde {M}}\rightarrow M}$

the universal overlay . This data creates a map (the so-called development map )

${\displaystyle dev:{\widetilde {M}}\rightarrow X}$

which, for each path, coincides with the analytical continuation along the path.

For differently selected output data and the development mapping changes only by the application of an element . ${\displaystyle x_{0}}$ ${\displaystyle \left(U_{0},\phi _{0}\right)}$ ${\displaystyle G}$

completeness

The development map is a local homeomorphism. A -manifold is called complete if its development map is surjective . If is simply connected , then every complete -manifold is of the form for a discrete subgroup . ${\displaystyle (G,X)}$ ${\displaystyle X}$ ${\displaystyle (G,X)}$ ${\displaystyle M=\Gamma \backslash X}$ ${\displaystyle \Gamma \subset G}$

There THAT CONDITION by analytic diffeomorphisms with compact stabilizers on . Then there is an -invariant Riemannian metric on every -manifold and the following conditions are equivalent: ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle (G,X)}$ ${\displaystyle M}$ ${\displaystyle G}$

• ${\displaystyle M}$is a full metric space .
• There is one such that all enclosed spheres are compact. ${\displaystyle \epsilon >0}$ ${\displaystyle \epsilon }$
• All completed spheres are compact.
• There is a family of compact sets with so that all the environment of in is included. ${\displaystyle \left\{S_{t}:t\in \mathbb {R} \right\}}$ ${\displaystyle M=\cup _{t}S_{t}}$ ${\displaystyle a,t}$ ${\displaystyle a}$ ${\displaystyle S_{t}}$ ${\displaystyle S_{t+a}}$

In particular, closed manifolds are always complete in this case . ${\displaystyle (G,X)}$

Holonomy

For

${\displaystyle \sigma \in \pi _{1}\left(M,x_{0}\right)}$

gives analytical continuation along a representative closed path a map comparable to , because both are defined on a subset of . Be ${\displaystyle \sigma }$ ${\displaystyle \phi _{0}}$ ${\displaystyle \phi _{0}^{\sigma }}$ ${\displaystyle U_{0}}$

${\displaystyle g_{\sigma }\in G}$,

so that

${\displaystyle \phi _{0}^{\sigma }=g_{\sigma }\phi _{0}}$.

The image

${\displaystyle H:\pi _{1}\left(M,x_{0}\right)\rightarrow G}$

${\displaystyle H\left(\sigma \right)=g_{\sigma }}$

is a group homomorphism and is called the holonomy of the structure. ${\displaystyle (G,X)}$

According to the construction, the development map is equivariant with respect to the holonomy homomorphism, i.e. H. it applies

${\displaystyle dev(\sigma x)=H(\sigma )dev(x)\ \forall x\in {\widetilde {M}},\sigma \in \pi _{1}(M,x_{0})}$.

For differently chosen initial data and the holonomy changes only to conjugation with an element . So you have a picture ${\displaystyle x_{0}}$ ${\displaystyle \left(U_{0},\phi _{0}\right)}$ ${\displaystyle G}$

${\displaystyle hol:\left\{(G,X)-{\mbox{Strukturen auf}}\ M\right\}\to Hom(\pi _{1}M,G)/conj}$.

Bundle interpretation (Ehresmann-Thurston-Weil theorem)

A structure on with (G, X) -Atlas and coordinate transitions can be a fiber bundle ${\displaystyle (G,X)}$ ${\displaystyle M}$ ${\displaystyle \left\{\left(U_{i},\phi _{i}\right)\right\}}$ ${\displaystyle \gamma _{ij}}$

${\displaystyle X\to E\to M}$

assign whose transition images are just the . In this interpretation, the development image corresponds to a cut . So the bundle is a flat bundle with monodromy . ${\displaystyle \gamma _{ij}}$ ${\displaystyle F\colon M\to E}$ ${\displaystyle E}$ ${\displaystyle H}$

Conversely, a section corresponds to a structure if it is transverse to the leaves defined by . ${\displaystyle F\colon M\to E}$ ${\displaystyle (G,X)}$ ${\displaystyle \left\{U_{i}\times \left\{x\right\}:x\in X\right\}}$

Because transversality is an open condition, it follows that there is a local homeomorphism. ${\displaystyle hol}$

Examples

Model geometries

A model geometry is a differentiable manifold with a more differentiated effect of a Lie group that satisfies the following conditions: ${\displaystyle X}$ ${\displaystyle G}$

• ${\displaystyle X}$ is connected and simply connected
• ${\displaystyle G}$acts transitively with compact stabilizers (in particular there is an -invariant Riemannian metric) ${\displaystyle X}$ ${\displaystyle G}$
• ${\displaystyle G}$is maximal among groups that act through diffeomorphisms with compact stabilizers ${\displaystyle X}$
• there is at least one compact manifold. ${\displaystyle (G,X)}$

From the last condition it follows in particular that it must be unimodular . There are numerous pairs that satisfy all but the last, for example , the Lie group of affine maps of the Euclidean plane. ${\displaystyle G}$ ${\displaystyle (G,X)}$ ${\displaystyle G=X=Aff(\mathbb {R} ^{2})}$

2-dimensional model geometries

2-dimensional model geometries were classified by Cartan, they are the 2-dimensional sphere, the Euclidean plane and the hyperbolic plane, each with their full isometric groups .

3-dimensional model geometries

→ Main article : "Model geometries" in the article Geometrization of 3-Manifolds

3-dimensional model geometries were classified by Thurston. There are eight 3-dimensional model geometries, where the isometric group of the homogeneous metric is: ${\displaystyle G}$

• the Euclidean space , ${\displaystyle \mathbb {R} ^{3}}$
• the three-dimensional sphere (surface of a four-dimensional sphere), ${\displaystyle S^{3}}$
• the hyperbolic space , ${\displaystyle \mathbb {H} ^{3}}$
• the product of 2-sphere and straight line , ${\displaystyle S^{2}\times \mathbb {R} }$
• the product of the hyperbolic plane and the straight line , ${\displaystyle H^{2}\times \mathbb {R} }$
• ${\displaystyle {\tilde {\mathrm {SL} }}(2,\mathbb {R} )}$, the universal superposition of the special linear group ${\displaystyle \mathrm {SL} (2,\mathbb {R} )}$
• the Heisenberg Group ${\displaystyle Nil}$
• the 3-dimensional resolvable Lie group . ${\displaystyle Sol}$

4-dimensional model geometries

4-dimensional model geometries were classified by Filipkiewicz.

Affine manifolds

Affine manifolds are -manifolds for and the group of affine maps. The Auslandser's Conjecture (proved by Fried and Goldman for n = 3) says that the fundamental group of compact affine manifolds is polycyclic . ${\displaystyle (G,X)}$ ${\displaystyle X=\mathbb {R} ^{n}}$ ${\displaystyle G}$

Conformal manifolds

A conforming structure is a structure with and . ${\displaystyle (G,X)}$ ${\displaystyle X=S^{n}=\partial _{\infty }H^{n}}$ ${\displaystyle G=O^{+}(n+1,1)}$

Projective manifolds

Projective manifolds are -manifolds for . In this case the structures correspond to the flat projective relationships . ${\displaystyle (G,X)}$ ${\displaystyle (G,X)=(PGL(n+1,\mathbb {R} ),\mathbb {R} P^{n})}$ ${\displaystyle (G,X)}$

Complex projective manifolds are -manifolds for . ${\displaystyle (G,X)}$ ${\displaystyle (G,X)=(PGL(n+1,\mathbb {C} ),\mathbb {C} P^{n})}$

Flag structure

A flag structure is a structure with and the flag manifold , i.e. H. the space of the complete flags in , with the canonical effect of and stabilizer the subset of the upper triangular matrices . ${\displaystyle (G,X)}$ ${\displaystyle G=GL(n,\mathbb {R} )}$ ${\displaystyle X=G/P}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle GL(n,\mathbb {R} )}$ ${\displaystyle P}$

Hierarchies of geometries

If a homomorphism and one - equivariant local diffeomorphism , then each is -manifold automatically a -manifold. ${\displaystyle \iota \colon G\to G^{\prime }}$ ${\displaystyle j\colon X\to X^{\prime }}$ ${\displaystyle \iota }$ ${\displaystyle (G,X)}$ ${\displaystyle (G^{\prime },X^{\prime })}$

For example, the Beltrami-Klein model of hyperbolic geometry shows that every hyperbolic manifold is automatically also a projective manifold . The other 3-dimensional Thurston geometries, with the exception of and, can also be interpreted as a subset of the projective geometry. ${\displaystyle S^{2}\times \mathbb {R} }$ ${\displaystyle H^{2}\times \mathbb {R} }$

literature

• William P. Thurston : Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. ISBN 0-691-08304-5
• G. Peter Scott : The geometries of 3-manifolds. Bull. London Math. Soc. 15 (1983) no. 5, 401-487. on-line
• Richard Canary ; David Epstein ; PL Green: Notes on notes of Thurston . With a new foreword by Canary. London Math. Soc. Lecture Note Ser., 328, Fundamentals of hyperbolic geometry: selected expositions, 1–115, Cambridge Univ. Press, Cambridge, 2006
• William P. Thurston: The Geometry and Topology of Three-Manifolds online
• Yoshinobu Kamishima; Ser Peow Tan: Deformation spaces on geometric structures. Aspects of low-dimensional manifolds, 263-299, Adv. Stud. Pure Math., 20, Kinokuniya, Tokyo, 1992
• William M. Goldman : Locally homogeneous geometric manifolds. Proceedings of the International Congress of Mathematicians. Volume II, 717-744, Hindustan Book Agency, New Delhi, 2010. pdf

Web links

• William P. Thurston: Geometric structures on manifolds
• Jonathan Hillman: 4-dimensional geometries
• Fanny Kassel: Geometric structures and representations of discrete groups
• William M. Goldman: WHAT IS ... a projective structure?
• Sam Ballas: Deformations and twisted cohomology

proof

1. ^ RP Filipkiewicz: Four-dimensional geometries , Ph.D. Thesis, Univ. Warwick, Coventry, 1984; per bibl.
2. ↑ CTC Wall : Geometries and geometric structures in real dimension 4 and complex dimension 2. Geometry and topology (College Park, Md., 1983/84), 268–292, Lecture Notes in Math., 1167, Springer, Berlin, 1985