Adapted complex structures and the geodesic flow
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Publication:540430
DOI10.1007/S00208-010-0564-9zbMATH Open1228.53097arXiv0811.3083OpenAlexW2128428804WikidataQ125260097 ScholiaQ125260097MaRDI QIDQ540430
Author name not available (Why is that?)
Publication date: 3 June 2011
Published in: (Search for Journal in Brave)
Abstract: In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the "complexifier" approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the "imaginary-time geodesic flow" to the vertical polarization. Meanwhile, at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the fibers by "composition with the imaginary-time geodesic flow." We give several equivalent interpretations of this composition, including a convergent power series in the vector field generating the geodesic flow.
Full work available at URL: https://arxiv.org/abs/0811.3083
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