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On the topological properties of Lagrangian projections in symplectic geometry of caustics. (Sur les propriétés topologiques des projections lagrangiennes en géométrie symplectique des caustiques.) (French) Zbl 0973.53501

The caustics of a point on a Riemannian manifold consist of the set of intersection points of infinitesimally closed geodesics starting at this point. Jacobi observed that the caustics of a point on a closed convex surface should have a cusp. He also announced that for the surface of the ellipsoid the number of cusps is equal to four (the author calls this “The last geometrical theorem of Jacobi”).
The author proves the four cusps theorem for caustics of the projection of the deformed Lagrangian cylinder to the plane in the framework of symplectic topology. This theorem is considered as a beautiful generalization of Sturm theory and is also closely related to the classical four vertices theorem on plane curves.

MSC:

53A04 Curves in Euclidean and related spaces
57M25 Knots and links in the \(3\)-sphere (MSC2010)
58K05 Critical points of functions and mappings on manifolds
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
53D05 Symplectic manifolds (general theory)
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