2011/03/28 by Aguilar, Raúl M.
#32q15 #32v40 #53c15 #53c20 #53c22 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1103.5305
In the first part of the paper, comprising section 1 through 6, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. With these functions and a classical theorem of Loewner on analytic continuation we are able to characterize the existence of the adapted complex structure induced by g on the set TRM of vectors in TM of length up to R, equivalently for M compact, to the existence of a Grauert tube of radius R in terms of infinite Hankel matrices involving these Schwarzian functions. The basic characterization so obtained can be expressed as a sequence of differential inequalities of increasing order polynomial in the covariant derivatives of the Gauss curvature on M and in π/R that should be regarded as the higher order versions of a curvature inequality by L. Lempert and R. Szöke. The second part of the paper, sections 7 through 11, includes a discussion of the rank of the infinite Hankel matrix of the Schwarzians from part 1 and of new Schwarzians defined now for purely imaginary radius, as well as some computations and examples. A characterization of the existence of the adapted structure on TRM in terms of moment sequences with parameters R and v in TM is also noted.