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In the first part of the paper, comprising section 1 through 6, we introducea sequence of functions in the tangent bundle TM of any smooth two-dimensionalmanifold M with smooth Riemannian metric g that correspond to the higher orderSchwarzians of the linearized geodesic flow. With these functions and aclassical theorem of Loewner on analytic continuation we are able tocharacterize the existence of the adapted complex structure induced by g on theset T^RM of vectors in TM of length up to R, equivalently for M compact, to theexistence of a Grauert tube of radius R in terms of infinite Hankel matricesinvolving these Schwarzian functions. The basic characterization so obtainedcan be expressed as a sequence of differential inequalities of increasing orderpolynomial in the covariant derivatives of the Gauss curvature on M and in{\pi}/R that should be regarded as the higher order versions of a curvatureinequality by L. Lempert and R. Sz\"oke. The second part of the paper, sections7 through 11, includes a discussion of the rank of the infinite Hankel matrixof the Schwarzians from part 1 and of new Schwarzians defined now for purelyimaginary radius, as well as some computations and examples. A characterizationof the existence of the adapted structure on T^RM in terms of moment sequenceswith parameters R and v in TM is also noted.

Higher Order Schwarzians for Geodesic Flows, Moment Sequences, and the Radius of Adapted Complexifications