An application of the $h$-principle to $C^1$-isometric immersions in contact manifolds

Abstract

We prove here a version of the Nash C 1 -isometric immersion theorem for contact manifolds equipped with Carnot-Caratheodory metrics. Introduction and motivation.Let us first recall the logical structure of the CMsometric theory for the Riemannian manifolds without contact structure. Here one starts with a general smooth (not at all isometric) immersion /o of a Riemannian manifoldIf V n is compact, then by an obvious scaling one can make such /o : (V 71 , g) -> R 9 strictly short. This means that the Riemannian metric induced by /o from R 9 is strictly smaller than g, i.e. the difference g -go is positive definite on V n . The key idea of the CMmmersion theory of Nash and Kuiper is as follows. One "stretches" a given strictly short immersion /o to an isometric CMmmersion /i, i.e. such that the form gi induced by /i equals g. This remarkable stretching was performed in the celebrated 1954 paper [13] by Nash under the assumption q > n + 2 and then Kuiper (1955) improved this result by showing that it is true when q -n + 1. (Clearly, this is impossible, in general, for q = n.)To complete the construction what remained was to have at one's disposal the starting immersion /o : V n -> R 9 . For this one could invoke the classical result by Whitney which claims that such an /o always exists for q > 2n. In fact, a generic C^-map / : V -> Il q is an immersion. Another possibility is offered by the Smale-Hirsch immersion theory which provides smooth immersions V n -> R g for a given q > n, provided the manifold V n satisfies the necessary topological restrictions. For example, every parallelizable (e.g. contractible) manifold V n can be smoothly immersed to 1 Work partially supported by the National Research Project: "Classificazione e proprieta geometriche delle varieta reali e complesse" (ex 40% Murst) 347