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