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A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to a geometric quantization. I. (English) Zbl 0830.53052

The authors consider the punctured cotangent bundle \(T^*_0 P^n\mathbb{C}\) (resp. \(T^*_0 P^n \mathbb{H})\) of the complex (resp. quaternion) projective space \(P^n\mathbb{C}\) (resp. \(P^n \mathbb{H}\)) and prove that the bundle space admits a Kählerian structure whose Kähler form coincides with the symplectic form, just like in the case of the spheres. The authors also describe the automorphisms of \(T^*_0 P^n\mathbb{C}\) and \(T^*_0 P^n H\). The arguments are based on the diagonalization of the geodesic flows.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53D50 Geometric quantization
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
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