Hypersurfaces in symplectic affine geometry. (Q1397831)

This paper is an introduction to symplectic affine geometry. Let \(f: M^{2n-1}\to \mathbb{R}^{2n}\) be an immersion. Denote by \(\Omega\) the standard symplectic form in \(\mathbb{R}^{2n}\). The immersion is called nondegenerate in the symplectic sense if the osculating 2-planes of the characteristic curves are transversal to the tangent plane of the immersion. The author shows that for a nondegenerate immersion there exists a unique vector field \(X_0\) along \(M^{2n-1}\), which generates the characteristic foliation and satisfies \(\Omega(f_* X_0, D_{X_0} f_* X_0)= 1\). The vector field \(\xi= D_{X_0} f_* X_0\) is called the symplectic normal vector field. \(\xi\) induces, in a natural way, a volume form on \(M^{2n-1}\). The author computes the first and second variation of the volume for spheres. He shows that spheres are strongly unstable: the index and co-index are both infinite.