Contact Equivalence for Lagrangian Manifolds M. GOLUBITSKY AND V. GUILLEMIN*

Let 2 be a manifold and X, Yr , Ya equidimensional submanifolds, all intersecting at 0 E X. Y, and Yz are said to be contact eqil&.~lent (r&A respect to X) at 0 if there exists a germ of diffeomorphism f: (2, 0) + (Z, 0) mapping X into X and Yr into Yz . The notion of contact equivalence is due to John Mather and plays an important role in his theory of singularities of differentiable mappings. This paper has to do with a slightly modified notion of contact equivalence, namely Z is assumed to be a symplectic manifold; X, Y, and Yz are assumed to be Lagrangian submanifolds, and f is assumed to be a germ of a symplectic diffeomorphism. Our main theorem (Proposition 3.2) states that two Lagrangian submanifolds have the same contact with a third if certain algebraic data of contact (a local ring and a distinguished element) are isomorphic. This is reminiscent of a theorem of Mather [6, Section 2.21 for ordinary contact equivalence which, for motivational purposes we describe in Section 2. The proof of Proposition 3.2 requires some results from symplectic geometry which we describe in Section 1. In Section 4 we exploit the fact that to each function 4 on a manifold X there is an associated Lagrangian submanifold, namely graph 4, in T*X to reformulate in symplectic form a theorem about right equivalence due to Tougeron [8, p. 2091. In the last section we give some examples to show that the algebraic criteria for contact equivalence given in Section 3 can not be weakened. To conclude we note that this paper had its origins in an attempt (unsuccessful) on our part to find a simple formula for the order of the caustic associated to a (germ of a) Lagrangian manifold (A, A) C T*X (see [l, Definition 1.6.11). Th e results here show that this number is a