Grassmann manifolds and the Grassmann image of submanifolds

Borisenko, A. A.; Nikolaevskii, Yu. A.

doi:10.1070/rm1991v046n02abeh002742

Cited by 34 publications

(36 citation statements)

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“…This embedding is defined by t.he Plficker coordinates; explicitly, to each oriented subspace that is the span of the vectors el, i = 1,..., r, we assign the point of E N with radius vector R = R1/[RI[, where R1 --[el, ..., er] is an oriented multivector. This Pliicker embedding is isometric and is equivalent to the one constructed in [7] .…”

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confidence: 76%

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

Горькавый¹

1996

Math Notes

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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.

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confidence: 76%

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

Горькавый¹

1996

Math Notes

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“…Indeed, conditions (B 1 )-(B 2 ) are independent of the dimension of the ambient space. As for condition (B 3 ), we should recall that the mutual arrangement of a pair of two-dimensional planes in E 4 is defined by two angles (stationary values of the angle between directions in these planes; see [6]); but since condition (B 3 ) is concerned with planes T P F 2 and TPF 2 , whose intersection is the straight line PP by virtue of condition (B 1 ), we have that the mutual arrangement of the planes T P F 2 and TPF 2 in E 4 is defined by the unique angle α, which is the angle between directions in T P F 2 and TPF 2 , orthogonal to the straight line PP , and precisely this angle α appears in condition (B 3 ) under the generalization of Definition 1.1 from E 3 to E 4 in [7].…”

Section: Theorem 22mentioning

confidence: 99%

Generalization of the Bianchi–Bäcklund Transformation of Pseudo-Spherical Surfaces

Горькавый

2015

J Math Sci

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“…. , θ k the principal angles between E and G as defined above; see [2] and the references given there. We shall always use this distinguished metric.…”

Section: Geometry Of Grassmann Manifoldsmentioning

confidence: 99%