By Maks A. Akivis, Vladislav V. Goldberg

During this ebook the authors learn the differential geometry of types with degenerate Gauss maps. They use the most tools of differential geometry, specifically, the equipment of relocating frames and external differential types in addition to tensor tools. via those equipment, the authors detect the constitution of sorts with degenerate Gauss maps, be sure the singular issues and singular forms, locate focal photos and build a category of the kinds with degenerate Gauss maps.The authors introduce the above pointed out tools and practice them to a sequence of concrete difficulties bobbing up within the concept of types with degenerate Gauss maps. What makes this e-book particular is the authors' use of a scientific software of equipment of projective differential geometry in addition to equipment of the classical algebraic geometry for learning kinds with degenerate Gauss maps. This ebook is meant for researchers and graduate scholars drawn to projective differential geometry and algebraic geometry and their purposes. it may be used as a textual content for complex undergraduate and graduate scholars. either authors have released over a hundred papers each one. every one has written a few books, together with Conformal Differential Geometry and Its Generalizations (Wiley 1996), Projective Differential Geometry of Submanifolds (North-Holland 1993), and Introductory Linear Algebra (Prentice-Hall 1972), that have been written by way of them together.

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58) where xijk = xikj . 57) are the ﬁrst set of structure equations of the manifold M n . 57) that the system of equations ω i = 0 is completely integrable. The ﬁrst integrals of this system are the coordinates xi of a point x of the manifold M n . 2 Diﬀerentiable Manifolds 17 Let us ﬁnd the second set of the structure equations of the manifold M n , which are satisﬁed by the forms ωji . 58) leads to the equations dωji = −dxkj ∧ dxik + dxijk ∧ ω k + xijk ω l ∧ ωlk . 59) The entries of the matrices (xji ) and (xji ) are connected by the relation xkj xik = δji .

50) deﬁning a vector ξk+1 , which is in involution with the previously deﬁned vectors ξ1 , . . , ξk , and let s1 = r1 , s2 = r2 − r1 , . . , sq−1 = rq−1 − rq−2 . 50) for ﬁnding a vector ξq . The integers s1 , s2 , . . 46), and the integer Q = s1 + 2s2 + . . 2 Diﬀerentiable Manifolds 15 is called its Cartan number. 46) are connected by the inequalities s1 ≥ s2 ≥ . . ≥ sq . 48) are exterior products of some linear forms from which q forms are linearly independent and are the basis forms of the integral manifold V q .

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]).