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On the stratification of the orbit space for the action of automorphisms on connections

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Kondracki W., Rogulski J.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Introduction..................................................................................................................................................5 §1. Basic notions and notation.....................................................................................................................7 1.1. Automorphisms of principal bundles....................................................................................................7 1.2. Connections and parallel translations.................................................................................................9 1.3. Symmetries and connections.............................................................................................................11 §2. The action of the gauge group on connections....................................................................................14 2.1. The gauge group...............................................................................................................................14 2.2. The action of on $G^{k+1}$ on $C^k$................................................................................................17 2.3. Weak and strong invariant metrics on $C^k$.....................................................................................20 2.4. The equivariant embedding of $C^k$ into the space of $H^k$ Riemannian metrics on P...................23 §3. The Slice Theorem...............................................................................................................................30 3.1. The Hodge-Kodaira-like decomposition for $T_e Φ_A$....................................................................30 3.2. The orbits are submanifolds..............................................................................................................36 3.3. The Slice Theorem............................................................................................................................38 §4. The geometric structure of $R^k = C^k/G^{k+1}$..................................................................................43 4.1. Consequences of the Slice Theorem.................................................................................................44 4.2. The Countability Theorem.................................................................................................................47 4.3. Density theorems...............................................................................................................................49 4.4. The stratification of $R^k$.................................................................................................................57 References.................................................................................................................................................61

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1 Kondracki W.

1 Rogulski J.

1 1986

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On the stratification of the orbit space for the action of automorphisms on connections