Ad -c.r. geometries in dimension less than/equal to 4.
Item
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Title
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Ad -c.r. geometries in dimension less than/equal to 4.
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Identifier
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AAI9009798
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identifier
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9009798
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Creator
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Wang, Ming.
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Contributor
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Adviser: Ravi S. Kulkarni
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Date
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1989
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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In this dissertation, I am studying a special kind of homogeneous geometries in dimension {dollar}\leq 4{dollar}. A geometry means a pair ({dollar}G,H{dollar}) where {dollar}G{dollar} is a Lie group, {dollar}H{dollar} its closed subgroup, both connected, such that {dollar}G{dollar} acts effectively on {dollar}S = G/H{dollar}, and {dollar}S{dollar} is simply connected. The dimension of the geometry is the dimension of {dollar}S{dollar}. The linear isotropy representation {dollar}\rho{dollar}: {dollar}H \rightarrow Aut(T\sb{lcub}x\sb{lcub}\rm o{rcub}{rcub}(S)), x\sb{lcub}\rm o{rcub} = \lbrack H\rbrack {dollar} being the basepoint of {dollar}S{dollar}, is closely related to the adjoint reepresentation of {dollar}H{dollar} on the Lie algebra {dollar}L(G){dollar} of {dollar}G{dollar}. We say that {dollar}(G,H){dollar} is {dollar}Ad - c.r.{dollar} if this representation is completely reducible and if {dollar}\rho(H){dollar} is closed in {dollar}Aut(T\sb{lcub}x\sb{lcub}\rm o{rcub}{rcub}(S)) \simeq GL\sb n(R), n = dim{dollar} {dollar}S{dollar}. In S1- S4 of this dissertation, {dollar}Ad-c.r.{dollar} geometries in dimension {dollar}\leq 4{dollar} are classified. The topology of each geometry is given in S5. In S6, for the most part of the classification list, their boundedness are determined (we call a geometry {dollar}(G,H){dollar} bounded, if there exists a subgroup {dollar}\Gamma{dollar} of G acting properly discontinuously on {dollar}S{dollar} such that {dollar}\Gamma/S{dollar} is compact). In S7, the flat complete compact pseudo- Riemannian space-forms with signature (2,2) is classified up to finite covers (This is the study of co-compact properly discontinuous subgroups in the geometry {dollar}(SO\sb{lcub}\rm o{rcub}(2,2) \times R\sp4, SO\sb{lcub}\rm o{rcub}(2,2))){dollar}. Our main theorem is: Suppose X is a flat compact complete space-form with fundamental group {dollar}\Gamma \subseteq SO\sb{lcub}\rm o{rcub}(2,2) \times R\sp4{dollar}, then there is a uniquely determined subgroup H of {dollar}SO\sb{lcub}\rm o{rcub}(2,2) \times R\sp4{dollar} that acts simply transitively on {dollar}R\sp4{dollar} and {dollar}H\cap \Gamma{dollar} has finite index in {dollar}\Gamma{dollar}. Then we need only to find the simply transitive subgroups and their uniform lattices. To prove the main theorem, we first proved that {dollar}\Gamma{dollar} is virtually solvable. This result confirms a conjecture by J. Milnor in a special case: The fundamental group of a complete affinely flat manifold is virtually polycyclic. In S8, {dollar}G{dollar}-invariant pseudo-Riemannian, contract, symplectic and complex structures on {dollar}G/H{dollar} in {dollar}dim. \leq 4{dollar} are classified.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
