On the affine Gauss maps of submanifolds of Euclidean space

TL;DR

This paper generalizes the concept of Gauss maps for submanifolds in Euclidean space to higher dimensions, introducing conditions for families of affine spaces and defining a curvature tensor that extends classical curvature notions.

Contribution

It extends the classical theory of Gauss maps and Lagrangian submanifolds to higher-dimensional affine families, including a new curvature tensor and a generalized Gauss-Bonnet formula.

Findings

Established conditions for the existence of submanifolds crossing affine spaces orthogonally.

Defined a curvature tensor generalizing classical submanifold curvature.

Proved a generalized Gauss-Bonnet formula for certain affine families in R^4.

Abstract

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.