  It is well known that the space of oriented lines of Euclidean space has a
natural symplectic structure. Moreover, given an immersed, oriented
hypersurface S the set of oriented lines that cross S orthogonally is a
Lagrangian submanifold. Conversely, if \bar{S} an n-dimensional family of
oriented lines is Lagrangian, there exists, locally, a 1-parameter family of
immersed, oriented, parallel hypersurfaces S_t whose tangent spaces cross
orthogonally the lines of \bar{S}. The purpose of this paper is to generalize
these facts to higher dimension: to any point x of a submanifold S of R^m of
dimension n and co-dimension k=m-n, we may associate the affine k-space normal
to S at x. Conversely, given an n-dimensional family \bar{S} of affine k-spaces
of R^m, we provide certain conditions granting the local existence of a family
of n-dimensional submanifolds S which cross orthogonally the affine k-spaces of
\bar{S}. We also define a curvature tensor for a general family of affine
spaces of R^m which generalizes the curvature of a submanifold, and, in the
case of a 2-dimensional family of 2-planes in R^4, show that it satisfies a
generalized Gauss-Bonnet formula.
