Adapted complex structures and the geodesic flow

Details Adapted complex structures and the geodesic flow Adapted complex structures and the geodesic flow

2008-11-19

arxiv.org/abs/0811.3083v2

Mathematics

Symplectic Geometry

13 pages, v2: minor corrections

Scientific paper

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.

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