Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the affine Gauss maps of submanifolds of Euclidean space   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.

37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas