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C is holomorphic if and only if = 0 for all j = 1, ... ,n. e. if and only if d¢ E n,p+l,q(M). This works in the same way for forms with values in any finite-dimensional complex vector space. 3. A survey on connections This section provides background on various versions of connections. We start from the simple idea of linear connections on vector bundles, then pass to general connections on fiber bundles, and specialize to principal and induced connections.

Consequently, we may specify the vertical projection of any general connection on P by a g-valued one form "I E [21(p,g), such that the vertical projection of ~ E TuP equals (1"(~)(u). e. "I ((x) = X for all X E g. One easily verifies that "I corresponds to a principal connection if and only if (r9)*"1 = Ad(g-I) 0 "I. We call the one-form "I the principal connection form, or briefly the principal connection, on P. The trivialization of the vertical bundle by fundamental vector fields also leads to a nice interpretation of the curvature of any connection on a principal fiber bundle.

X = {g. x : 9 E G} through x, and second there is the isotropy subgroup G x = {g E G : g . x = x}, also called the stabilizer of x. By definition, G x is a closed subgroup and thus a Lie subgroup in G. The map fX : G -+ M, fX(g) := f(g, x) then induces a smooth bijection GIG x -+ G . x, so any orbit looks like a coset space. Clearly, two orbits are either disjoint or equal, so M is the disjoint union of all G-orbits. The set of all orbits is denoted by MIG. Note that for y = g. x E G· x, one has G y = {ghg- 1 : h E G x }, so along an orbit all isotropy subgroups are conjugate.

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Parabolic Geometries I: Background and general theory by Andreas Čap, Jan Slovák


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