A few affine invariant structures depending only on the second fundamental form relative to arbitrary transversal bundles on submanifolds of the standard affine spaces are introduced. A notion of “local strong convexity” is proposed for arbitrary codimensional submanifolds. In the case of $n$-dimensional submanifolds of $2n$-dimensional real affine spaces, complex structures on the ambient spaces are used as a tool for studying real affine invariants.
"Weak geometric structures on submanifolds of affine spaces." Tohoku Math. J. (2) 60 (3) 383 - 401, 2008. https://doi.org/10.2748/tmj/1223057735