Abstracts

Synthetic or qualitative methods can provide insight into geometric structures not apparent from the purely analytic point of view. For example, much of the geometry of a classical surface in Euclidean space is coded in the differential topology of its family of normal lines. This thesis extends the range of applicability of such methods by developing the basic theory of submanifold families in general, emphasizing higher-order contact phenomena and making use of the modern theory of jets. A main technical contribution is, in certain cases, the calculation of invariants of parameterized submanifold jets by the reparameterization group, a construction of the quotient varieties, and compactifications of these varieties. As applications, in 2 dimensions we deduce: - a description of projective structures by (second order) tensorial data, - a characterization of the curve families realizable by geodesics for some connection, and a description of the connections in this case, and - a linearizability criterion for curve families, including d-webs for d > 3, and in higher dimensions: - general formulas for envelopes of submanifold families, including line envelopes for visual applications, and - a characterization of the extrinsic geometry of m-submanifolds in projective space RPn, for certain m, including a generalization of the classical Wilczynski equations for surfaces in RP3.