Constant Principal Strain Mappings On 2-Manifolds

We study mappings between Riemannian 2-manifolds which have constant principal stretching factors (cps-mappings). Such mappings f can be described in terms of the relationship between the geodesic curvature of the curves of principal strain at p and that of their images at f(p). In the context of local coordinates this relationship takes the form of a nonlinear hyperbolic system, the blow-up properties of which depend on the Gaussian curvatures of the two manifolds. We use the theory of such systems to study global existence when both manifolds are the hyperbolic plane $\Bbb{H}^2$ and obtain a simple description of all cps-mappings of $\Bbb{H}^2$ onto itself. We also obtain a distortion result for disks in $\Bbb{H}^2$ as well as some nonexistence results for cps-mappings of the Euclidean plane onto certain classes of manifolds. In addition, our treatment of cps-mappings in $\Bbb{H}^2$ yields, virtually as a corollary, a generalization of a theorem of Epstein to the effect that a curve in hyperbolic n-spac...