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On the Affine Gauss Maps of Submanifolds of Euclidean Space

par Anciaux, Henri ;Bayard, Pierre
Référence Bulletin Brazilian Mathematical Society, 50, 1, page (137-165)
Publication Publié, 2019-03

Article révisé par les pairs

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Résumé :

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 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 S¯. The purpose of this paper is to generalize these facts to higher dimension: to any point x of a submanifold S of Rm 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 S¯ of affine k-spaces of Rm, we provide certain conditions granting the local existence of a family of n-dimensional submanifolds S which cross orthogonally the affine k-spaces of S¯. We also define a curvature tensor for a general family of affine spaces of Rm which generalizes the curvature of a submanifold, and, in the case of a 2-dimensional family of 2-planes in R4, show that it satisfies a generalized Gauss–Bonnet formula.