Isotropic submanifolds of TS^ n and their focal sets

Families of oriented lines in Rⁿ⁺¹ are studied via their identification with submanifolds of TSⁿ . In particular, families of oriented lines which are isotropic with respect to the canonical symplectic structure on TSⁿ are shown to characterise those which are orthogonal to submanifolds of Rⁿ⁺¹ . We then study families of lines which are tangent to a k -dimensional submanifold of Rⁿ⁺¹ . For such families, isotropy is shown to be equivalent to an associated vector field on the submanifold being geodesic and hypersurface-orthogonal. The focal set in Rⁿ⁺¹ of a family of lines is introduced, extending the classical definition for families normal to hypersurfaces, to general families of lines of arbitrary codimension. A formula is derived that expresses certain sectional curvatures of the focal set in terms of the signed distances between corresponding focal points. We then solve an inverse problem for the focal sets of hypersurfaces and show certain sectional and Ricci curvatures of the focal set are determined by the differences between the hypersurface’s radii of curvature. This generalises a theorem of Bianchi from 1874 - namely that surfaces in R³ of constant astigmatism have pseudo-spherical focal sets.