On hyperbolic surfaces: Moving triad and Loewner system connections

The connection between the motion of certain curves in 3 and (1 + 1)-dimensional soliton equations is well-established. Here, moving orthonormal {t, n, b} triads are associated with integrable systems linked to hyperbolic surfaces as recently described by Levi and Sym [D. Levi and A. Sym. Integrable systems describing surfaces of nonconstant curvature, Phys. Lett. A 149, 381–387 (1990).] It is shown, in turn, that these geometrically based integrable systems arise as a subcase of the Loewner-Konopelchenko-Rogers (LKR) integrable systems. In particular, the latter is shown to generate novel Ernst-type equations.