Prescribing invariants for integral surfaces in the Grassmann bundle of 2-planes in 4-space

Abstract

A SMOOTHLY immersed integral surface M in the Grassmann bundle M > G,(R4) > R4 defines a (possibly nonimmersed) surface at 0 s(M) in R4. Moreover, a surface in R4 whose nonimmersive singularities are sufficiently well behaved, lifts to an integral surface in G,(R4). (This lift is defined by assigning a “tangent” 2-plane to each point on the surface.) Thus a compact integral surface in G,(R4) can be thought of as a resolution or lift of a nonimmersed surface in R4 which is well defined up to smooth diffeomorphisms of R4 and M. In this paper we will consider the set of compact orientable rank 1 transverse integral surfaces in G,(R4) [4]. We then define five differential topological invariants for such a surface (Section 2). In Theorem 1 we show that these invariants are independent and can be freely prescribed. As a consequence, any 2-plane bundle can be realized as the pull-back bundle s* (6) over a rank 1 transverse integral surface, where & is the tautological2-plane bundle over G2(R4) (Corollary 2). The proof of Theorem 1 is constructive and builds the desired integral surface by cutting and pasting the local models described in Section 3.2. Integral surfaces occur naturally as (multivalued) solutions to PDE [4]. Since a PDE can be viewed as a constraining submanifold in G2(R4) the invariants of an integral surface solution should be restricted by the PDE. In Section 5 (Theorem 3) we briefly describe such restrictions in the case where the PDE is type-homotopic to a determined linear elliptic or hyperbolic system on the plane (cf. [2, 3, 51).