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 
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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     author = {V. A. Gorkavyy},
     title = {Reconstruction of a~submanifold of {Euclidean} space from its {Grassmannian} image that degenerates into a~line},
     journal = {Matemati\v{c}eskie zametki},
     pages = {681--691},
     year = {1996},
     volume = {59},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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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