Defining Lagrangian immersions by phase functions

Lees, J. Alexander

doi:10.1090/S0002-9947-1979-0530051-1

Defining Lagrangian immersions by phase functions
HTML articles powered by AMS MathViewer

by J. Alexander Lees PDF

Trans. Amer. Math. Soc. 250 (1979), 213-222 Request permission

Abstract:

In order to analyze the singularities of the solutions of certain partial differential equations, Hörmander, in his paper on Fourier integral operators, extends the method of stationary phase by introducing the class of nondegenerate phase functions. Each phase function, in turn, defines a lagrangian submanifold of the cotangent bundle of the manifold which is the domain of the corresponding differential operator. Given a lagrangian submanifold of a cotangent bundle, when is it globally defined by a nondegenerate phase function? A necessary and sufficient condition is here found to be the vanishing of two topological obstructions; one in the cohomology and the other in the k-theory of the given lagrangian submanifold.

References

V. I. Arnol′d, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Priložen. 1 (1967), 1–14 (Russian). MR 0211415, DOI 10.1007/BF01075861
Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052
André Haefliger and Valentin Poenaru, La classification des immersions combinatoires, Inst. Hautes Études Sci. Publ. Math. 23 (1964), 75–91 (French). MR 172296, DOI 10.1007/BF02684311
J. Alexander Lees, On the classification of Lagrange immersions, Duke Math. J. 43 (1976), no. 2, 217–224. MR 410764

Morse theory

Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258, DOI 10.1515/9781400883875
T. E. Stewart, On groups of diffeomorphisms, Proc. Amer. Math. Soc. 11 (1960), 559–563. MR 120651, DOI 10.1090/S0002-9939-1960-0120651-6
Alan Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971), 329–346 (1971). MR 286137, DOI 10.1016/0001-8708(71)90020-X
Alan Weinstein, Lectures on symplectic manifolds, Regional Conference Series in Mathematics, No. 29, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976. MR 0464312, DOI 10.1090/cbms/029
Alan Weinstein, Singularities of families of functions, Differentialgeometrie im Grossen (Ber. Tagung, Math. Forschungsinst., Oberwolfach, 1969) Ber. Math. Forschungsinst. Oberwolfach, Heft 4, Bibliographisches Inst., Mannheim, 1971, pp. 323–330. MR 0423403