By Maks A. Akivis, Vladislav V. Goldberg
This booklet surveys the differential geometry of sorts with degenerate Gauss maps, utilizing relocating frames and external differential types in addition to tensor equipment. The authors illustrate the constitution of types with degenerate Gauss maps, verify the singular issues and singular types, locate focal photos and build a category of the types with degenerate Gauss maps.
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Extra resources for Differential Geometry of Varieties with Degenerate Gauss Maps (CMS Books in Mathematics)
Taking φ5 = 1 E5 , we arrive at b5 = 1. 146) that ω00 = b6 ω01 . 148), we obtain the exterior quadratic equation (db6 + ω10 ) ∧ ω01 = 0, which by Cartan’s lemma implies that db6 + ω10 = kω01 . 149) that δb6 = −π10 . Fixing all secondary parameters except the parameter φ6 , in terms of the differential of which the secondary form −π10 is expressed, we obtain consecutively −π10 = δφ6 , δb6 = δφ6 , b6 = φ6 + E6 , 38 1. Foundational Material where E6 = const. Taking φ6 = −E6 , we arrive at b6 = 0. 134) give ω00 = ω22 = 0.
67) can be written in the form i ωk ∧ ωl . 68) A more detailed presentation of the foundations of the theory of aﬃne connections can be found in the books [KN 63] by Kobayashi and Nomizu and [Lich 55] by Lichnerowicz (see also the papers [Lap 66, 69] by Laptev). 1 Projective Transformations, Projective Frames, and the Structure Equations of a Projective Space. We assume that the reader is familiar with the notions of the projective plane and the three-dimensional projective space. These notions can be generalized for the multidimensional case in a natural way (see Dieudonn´e [Di 64]).
In homogeneous coordinates, the equation of a conic in the plane P2 is a11 (x1 )2 + 2a12 x1 x2 + a22 (x2 )2 + 2a10 x1 x0 + 2a20 x2 x0 + a00 (x0 )2 = 0. It has six coeﬃcients but only ﬁve essential parameters. Thus a conic can have a fourth-order tangency with the curve Γ. We write the preceding equation in nonhomogeneous coordinates x and y as follows: a11 x2 + 2a12 xy + a22 y 2 + 2a10 x + 2a20 y + a00 = 0. 123), we must obtain ﬁve roots x = 0. 4 Specializations of Moving Frames 33 coeﬃcients in xk , k = 0, 1, 2, 3, 4, must vanish.