Algebraic geometry and local differential geometry

Abstract

In this paper we shall study the local differential geometry of a manifold in a linear space.We are interested in determining the structure of submanifolds whose position in the linear space fails to be generic in one of five specific ways ( 2 ).Algebraic geometry enters the problem quite naturally when we observe that the structure of sucessive infinitesimal neighborhoods of a point p in the manifold is described by a sequence of linear systems ( 3 ) in the tangent spaces Tp(M) constituting the 1st, 2nd, 3rd, 4th, .. .fundamental forms.These linear systems vary with p in a prescribed manner.Non-genericity turns out to be described by very strong pointwise implications on these linear systems.For fixed p these implications may be studied by algebro-geometric methods (Bertini's Theorem, Bezout's Theorem, elementary properties of base loci of rational maps, etc.).Then when p varies these algebro-geometric conclusions must fit together i.e. must satisfy integrability conditions, which leads to still further restrictions, and so forth.In more detail, we are primarily interested in the local differential geometry of a complexanalytic submanifold of projective space, written (0.1) McPâ nd with the full projective group as group of symmetries ( 4 ).On a few occasions we shall also be concerned with submanifolds of C N having either the affine linear or affine unitary group as symmetry group.In a neighborhood of a generic point on M ( 5 ), we may attach to each point p e M a sequence of linear systems |II|, |III|, |IV|, ... consisting of quadrics, cubics, quartics, ... in the projectivized tangent spaces PT^M)^?"" 1 where dimM=n.These linear systems constitute the 2nd, 3rd,( 2 ) These are discussed individually in paragraphs 2-6.( 3 ) Here we mean linear systems of divisors on projective space.In Appendix A we have collected the relevant definitions and facts from algebraic geometry.( 4 ) Aside from the results of paragraph 4 everything we shall do holds for real analytic submanifolds of W^.Also, projective space may be replaced by any manifold having a flat projective connection, so that e. g. our results are meaningful for submanifolds of non-Euclidean spaces.( 5 ) More precisely, we should be in an open set where a finite number of holomorphically varying matrices have constant maximal rank.