On encryption of infinitesimal neighbourhoods in geometric invariants of the conic structure on the space of nearby submanifolds.

摘要

The object of study here is an integrable conic connection defined on a general conic structure. This holomorphic second-order geometric structure was introduced in (13) and shown to be (naturally) equivalent to a double fibration inducing a holomorphic family of submanifolds; in this correspondence the underlying manifold of the geometric structure is the parameter space of the family. The following problems are considered: characterization of those conic structures which are induced by families of submanifolds, examination of the 'degree of reconstructibility' of the family from the conic structure alone, construction of an apparatus for translation of the invariants of an embedding into differential invariants of the induced conic structure, introduction of analogous invariants (namely fattenings of certain manifolds) even in the case of conic structures not induced by families of submanifolds, construction of distinguished connections etc. In connection with the above translation problem, we restrict our attention to the case of infinitesimal neighbourhoods of low order, but the method we develop seems to constitute the rudiments of a general approach to such problems more direct than the method used in (6). Furthermore, we obtain a generalization and reinterpretation in the context of conic structures of some of the results from (14) on locally complete parameter spaces of Legendrian submanifolds. Finally, the above general results are applied to the 'hypersurface-directional' conic structures equivalent to {dollar}Gsb{lcub}n{rcub}{dollar}-structures (in terminology of (3); the indicated structural group is a quotient of GL(2)). Among these applications are a generalization to arbitrary n of a theorem from (3)) on the spaces of Legendrian rational curves and {dollar}Gsb{lcub}n{rcub}{dollar}-structures, the description of a family of rational curves in a surface in terms of mutually compatible {dollar}Gsb{lcub}n{rcub}{dollar}-structure and projective structure, and determination of the values of the self-intersection index n for which the {dollar}Gsb{lcub}n{rcub}{dollar}-structure alone (subject to certain restrictions) suffices for that purpose. (These results generalize from the cases n = 1,2 to the general case the description of such families in (9).) Furthermore, we study the relationship between the intrinsic torsion (or torsion of the Cartan connection when the latter is defined) of a {dollar}Gsb{lcub}n{rcub}{dollar}-structure and the infinitesimal neighbourhoods of the rational curves. Apart from the theory of {dollar}Gsb{lcub}n{rcub}{dollar}-structures, we also show how some simple conic-structural invariants provide a tool for proving a result stated in (16) involving the first-order infinitesimal neighbourhood of an anti-self-dual Kaehler surface in its twistor space.