Extensions of flat partial connections

摘要

It is proved that any flat partial connection ▽~F on a line bundle L over a manifold M, where F is a complex integrable subbundle of (TM)~C, admits an extension to a connection ▽ on L. The kernel of the curvature form ω of ▽ is investigated. These results are applied to description of the Bohr-Sommerfeld subset in geometric quantization theory and quantization of the motion in the potential λ~2x~2 + μ~2/x~2.