Reconstruction of a submanifold of Euclidean space from its Grassmannian image that degenerates into a line

Abstract

ABSTRACT. We study the existence of a submanifold F" of Euclidean space E a+P with prescribed Grassmannian image that degenerates into a line. We prove that F is the Gra.ssmannian image of a regular submanifold F" of Euclidean space E '~+1' if and only if the curve r in the Grassmann manifold G+(p, n % p) is asymptotically Cr-regular, r > 1 Here G + (n, n + p) is embedded into the sphere S N , N = Cnr+p --("+v~ by the 9 x p i T Plfieker coordinates.w Let F n C E n+r be an oriented regular n-dimensional submanifold of (n + r)-dimensional Euclidean space. To each point Q E F" assign the oriented subspace E r C E "+r that is parallel to the oriented normal space NqF '~ of the submanifold F '~ at the point Q and carries the same orientation. This correspondence defines a map G from F" into the Grassmannian manifold G+(r, n + r) of oriented r-dimensional subspaces of (n + r)-dimensional Euclidean space. The image G(F") is called the Gras~-mannian image of the submanifold F" C E "+r. Let us denote it by r. In the general case the dimension of r is equal to the dimension of F".In [1] Aminov stated the following general problem: for a given regular submanifold r C G+(r, n + r) find a regular submanifold F" C E n+r whose Grassm~nrdan image is r". This problem was most fruitfully studied by Aminov himself (see [1,2]) and by Ho~man and Osserman (see [3]) for two-dimensional submanifolcls.Sometimes the dimension of the Grassmannian image r may be less than the dimension of the surface F '~ . This happens if the normal space of F" in E "+r is stationary along some submanifolds of F n . In this case the Grassmannian image is called degenerate and the submanifold F" is called tangentially degenerate. Tangentially degenerate surfaces are strongly parabolic in the sense of [4]. Aminov and Tarasova studied the reconstruction of the submanifold F n C E n+r from its degenerate Grassmannian image. They proved the following theorem.Theorem [5]. A curve r c G+(2, 4) is the degenerate Grassmannian image of a certain surface Here we prove the following general statement.Theorem. 1. /s a regular curve F in a Grassmann manifold G+(r, n % r) is the Grassmannian image of a CV-regular (p > 2) submanifold F n C E "+r, then r is an asymptotic curve os the manifold G+(r, n + r) C S N-x.2. Let F be a UP-regular (p > 1 ) asymptotic curve in G+(r, n + r) C S N-~ 9 Then/'or each point P E r there exist a neighborhood U C r and a Cp.+l-regular submanifold F" C E "+" such that the Grassmannian image of F '~ is U.Here and subsequently we consider the embedding of G+(r, n + r) into the unit sphere S Nz'l (N = Cnr+r, where Cnr+r here and below denotes the binomial coefficient ("+~)) defined by the Pliicker coordinates. Let us recall that a line 1" C G+(r, n + r ~) is called a.s~/mptotic for G+(r, n + r) if at any point Q E r the second fundamental form G+(r, n + r) C S N-I with respect to any normal is equal to zero in any direction tangent to r.

On the geometric results of A. V. Pogorelov

“…This problem was solved by Yu. A. Aminov [12,13], J. L. Weiner [14], and V. A. Gor'kavyi [15]. The existence and uniqueness theorem for two-dimensional surfaces was proved by Aminov.…”

Reconstruction of 3-submanifolds of large codimension in euclidean spaces from their gauss image

“…A. Aminov in [1], and it has attracted some attention. However, up to now it is solved only for two-dimensional regular submanifolds of n-dimensional Euclidean space with regular two-dimensional Gauss image (see the survey [2]), for regular submanifolds F '~ C Em+" , n, m > 1 with degenerate smooth one-dimensional Gauss image F 1 C G(m, m + n) [3], [4], and for regular submanifolds F n C E '~+2 , n > 1, with degenerate regular two-dimensional Gauss image F 2 C G(2, n + 2) [5]. We present some necessary and sufficient conditions for a regular three-dimensional manifold F ~ C G(m, m+3) to be the Gauss image of a submanifold F 3 C E m+3 9…”