Geodesic
Reconstruction of a~submanifold of Euclidean space from its Grassmannian image that degenerates into a~line
Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 681-691.

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We study the existence of a submanifold $F^n$ of Euclidean space $E^{n+p}$ with prescribed Grassmannian image that degenerates into a line. We prove that $\Gamma$ is the Grassmannian image of a regular submanifold $F^n$ of Euclidean space $E^{n+p}$ if and only if the curve $\Gamma$ in the Grassmann manifold $G^+(p,n+p)$ is asymptotically $C^r$-regular, $r>1$. Here $G^+(p,n+p)$ is embedded into the sphere $S^N$, $N=C_{n+p}^p$, by the Plücker coordinates. 
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     author = {V. A. Gorkavyy},
     title = {Reconstruction of a~submanifold of {Euclidean} space from its {Grassmannian} image that degenerates into a~line},
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V. A. Gorkavyy. Reconstruction of a~submanifold of Euclidean space from its Grassmannian image that degenerates into a~line. Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 681-691. http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a3/

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