Extensions of flat partial connections. (Q1573627)

Flat partial connections on a complex line bundle \(L\) as the objects of Kostant's geometric quantization theory were studied by \textit{K. Gawedzki} [Dissertationes Math., Warszawa 128, 78 p. (1976; Zbl 0343.53024)], \textit{J. H. Rawnsley} [Trans. Am. Math. Soc. 230, 235--255 (1977; Zbl 0313.58016); Proc. Am. Math. Soc. 73, 391--397 (1979; Zbl 0409.58004)] and others. Let F be a complex involutive subbundle of the complexified tangent bundle of a differentiable manifold \(M\). The author proves that any flat partial \(F\)-connection on a line bundle \(L\) over \(M\) admits an extension to a linear connection on \(L\). The kernel of the curvature of this linear connection is studied. Then the author applies his results to the description of the Bohr-Sommerfeld subset in geometric quantizations theory [see \textit{J. Sniatycki}, Geometric quantization and quantum mechanics. Applied Mathematical Sciences 30, New York: Springer (1980; Zbl 0429.58007)].