Surfaces with maximal curvature of the Grassmann image

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

doi:10.1007/bf01157428

Cited by 2 publications

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“…The study of the Grassmann image was conducted from such a point of view, for example, in [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

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On surfaces whose Grassmann image has curvature of at least one

Nikolaevskii¹

1991

J Math Sci

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CITED I.A.I. Medyanik, "Uniqueness theorems for regular closed convex surfaces," Ukr. Geom. Sb., No. 21, 86-88 (1983). 2.A.I. Medyanik, "The solvability of one equation for closed convex surfaces," Ukr. Geom. Sb., No. 31, 31-34 (1988). 3. A.I. Medyanik, "Reducibility of the general problem of existence for closed convex surfaces to C. Mirana's equation," Ukr. Geom. Sb., No. 29, 103-112 (1986). 4. A.V. Pogorelov, The Extrinsic Geometry of Convex Surfaces [in Russian], Moscow (1969). ON SURFACES WHOSE GRASSMANN IMAGE HAS CURVATURE OF AT LEAST ONE Yu. A. Nikolaevskii UDC 514We study the curvature of a Grassmann manifold along area elements tangent to a nondegenerate Grassmann image of a regular surface.According to Wong's theorem, it is included within the limits of [0; 2]. Limiting cases were considered earlier by Muto, Borisenko, and Nikolaevskii.There exists a conjecture according to which the values of the described curvatures cannot all be greater than I for surfaces of dimension at least 3. If they are all greater than or equal to i, then the surface is a hypersurface.The conjecture is proven for certain bounds from below on the dimension of the surface.

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“…The study of the Grassmann image was conducted from such a point of view, for example, in [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

“…Specifically, in [3,4] the limiting cases of the values of the sectional curvature of the Grassmann manifold along tangent areaelements to the Grassmann image were studied and, in partiuclar, the following were proved: THEOREM A [3]. Let Ft~E I+p be a regular, class C 2 surface with a nondegenerate Grassmann image.…”

Section: Introductionmentioning

confidence: 99%