Classifying spaces for homogeneous manifolds and their related Lie isometry deformations

Details Classifying spaces for homogeneous manifolds and their related Lie isometry deformations Classifying spaces for homogeneous manifolds and their related Lie isometry deformations

1996-02-28

arxiv.org/abs/gr-qc/9602059v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Latex, 22 pages, 3 figures available as hardcopy

Scientific paper

Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their continuous deformations is presented: Classifying spaces for homogeneous manifolds and their related Lie isometry deformations. The adjoint representation of n-dimensional real Lie algebras induces a natural topology on their classifying space, which encodes the natural algebraic relationship between different Lie algebras therein. For n>1 this topology is not Hausdorffian. Even more it satisfies only the separation axiom T_0, but not T_1, i.e. there is a constant sequence which has a limit different from the members of the sequence. Such a limit is called a transition. Recently it was found that transitions are the natural generalization and transitive completion of the well-known In\"on\"u-Wigner contractions. For n<5 the relational classifying spaces are constructed explicitly. Calculating their characteristic scalar invariants via triad representations of the characteristic isometry, local homogeneous Riemannian 3-spaces are classified in their natural geometrical relations to each other. Their classifying space is a composition of pieces with different isometry types. Although it is Hausdorffian, different topological transitions to the same limit may induce locally non-Euclidean regions (e.g. at Bianchi tppes VII_0).

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