Weak geometric structures on submanifolds of affine spaces (Q957508)

For submanifolds of the affine space affine invariants are studied, using only the second fundamental form \(h\) determined by any transversal bundle \({\mathcal N}\). Four equivalent conditions are given which characterize the local strong convexity of the submanifold \(f:M\to \mathbb{R}^N\) at \(x_0\in M\), i.e., the existence of a vector hyperplane \(\Omega_{x_0}\) of transversal bundle \({\mathcal N}_{x_0}\) such that all vectors \(h(X,X)\) for \(X\in T_{x_0}M\), \(X\neq 0\), lie on one side of \(\Omega_{x_0}\). A special attention is paid to \(n\)-dimensional submanifolds of the affine space \(\mathbb{R}^{2n}\), every of which can be locally viewed as a purely real (in other words affine Lagragian) submanifold relative to some complex structure on \(\mathbb{R}^{2n}\). A local classification is given of surfaces in \(\mathbb{R}^4\) for which the conformal structure vanishes. Also, a description is given of compact orientable surfaces in \(\mathbb{R}^4\) whose rank of the conformal structure is constant.