The geometry of immersions. II

Abstract

We hereby announce some general methods for higher order differ­ ential geometry. Our main tools are a generalization of the general position theory of Whitney and Thorn, and a characteristic class theory for higher order bundles having given higher order connec­ tions. The second part of this announcement will deal with applications of this machinery to some problems of geometric singularities. One application will be to count the number of umbilic points on an im­ mersed hypersurface. Full details will appear in a separate publica­ tion. 1. pth order osculating maps. It is known [4] that on each smooth manifold X a sequence of smooth vector bundles { Tk(X)} k-ix... over X can be canonically constructed. TP(X) is called the bundle of pth order tangent vectors over X, and Ti{X) is just the tangent bundle of X. These bundles furthermore satisfy short exact sequences 0 -> Tr-i(X) -> TP(X) -> OTitX) -> 0 where OpTi(X) denotes the p-iold symmetric tensor product of the tangent bundle. It is also known that to each smooth map ƒ between manifolds X and Y there is canonically defined a pth order differential Tp(f). TP(X)-*TP(Y) which is a homomorphism of smooth vector bundles covering ƒ. For each smooth ƒ there is the following family of commutative diagrams of vector bundles with exact rows, 0 > TP^(X) • TP(X) • 0*Ti(X) > 0