On certain canonical structures on tangent bundles of Riemannian manifolds.

In the first part of this Dissertation we define and study certain almost complex structures (a.c.s.) in tangent bundles of Riemannian manifolds, which we call isotropic and are integrable, i.e., complex structures, precisely when ${\cal M}$ has constant sectional curvatures.(cf. (1)) We characterize those isotropic integrable a.c.s. that are invariant by the full isometry group of ${\cal M}$, acting by tangent maps, in terms of a two parameter family of real functions of the energy function. We show that in the tangent bundle of hyperbolic space, there are isotropic integrable almost complex structures that are not invariant in the above sense. In the second part of this dissertation we define the real analogue of the Lempert-Szoke-Guillemin-Stenzel canonical adapted complex structure on tubular neighborhoods of the zero section in the tangent bundle of a real analytic Riemannian manifold introduced in (2) (the Grauert tube of ${\cal M}$.) This dual is the adapted product structure and is geometrically interpreted in terms of the shape operator of geodesic spheres. We define a CR analogue of the Anosov splitting, as a limit of a rescaled version of the complex structure. This CR limit, when it exists, is a geodesic flow invariant CR structure in the unit tangent bundle of ${\cal M}$. We show that among symmetric spaces of compact type and a certain family of metrics of revolution on the two sphere, such CR limit exists precisely for the rank symmetric cases and relate this to the thickening of the Gauss map for such metrics of revolution. We generalize, as follows, Stenzel's reduction of the equation for a Ricci-flat Kahler metric in the Grauert tubes of rank 1 symmetric spaces of compact type to an ODE found in (4). We establish a real analogue of a Kahler metric where the complex structure is replaced by a product structure P, and a non-degenerate symmetric two tensor, anti-invariant by P. The equation Einstein pseudo-Riemannian metrics on tubes containing the zero section in the tangent bundle reduces to an ODE when ${\cal M}$ is an harmonic manifold. Upon complexification, the equation for a Kahler-Einstein metric on the Grauert tubes of harmonic Riemannian manifolds reduces to an ODE and we study the completeness of the metrics obtained.PhDMathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/130642/2/9811017.pd