By Maks A. Akivis, Vladislav V. Goldberg

ISBN-10: 0387404635

ISBN-13: 9780387404639

During this publication the authors research the differential geometry of types with degenerate Gauss maps. They use the most tools of differential geometry, particularly, the tools of relocating frames and external differential varieties in addition to tensor tools. by way of those equipment, the authors notice the constitution of types with degenerate Gauss maps, be sure the singular issues and singular types, locate focal pictures and build a class of the types with degenerate Gauss maps.

The authors introduce the above pointed out equipment and follow them to a chain of concrete difficulties coming up within the thought of types with degenerate Gauss maps. What makes this ebook certain 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 types with degenerate Gauss maps.

This e-book is meant for researchers and graduate scholars attracted to projective differential geometry and algebraic geometry and their purposes. it may be used as a textual content for complicated undergraduate and graduate scholars.

Both authors have released over a hundred papers each one. each 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 via them together.

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**Sample text**

N. Because the center Pm is unchanged under projectivization, the equations of inﬁnitesimal displacement of the frame {Ai , Aα } of the space Pn can be written in the form dAi = ωij Aj , dAα = ωαβ Aβ + ωαi Ai . Thus, in this family of frames we have ωiα = 0. 73) of a projective space Pn imply that dωαβ = ωαγ ∧ ωγβ . 80) This allows us to consider the forms ωαβ as the components of inﬁnitesimal displacement of the frame {Aα } of the projectivization Pn−m−1 , so that dAα = ωαβ Aβ . 3 Projective Space 25 On some occasions, we will identify the points Aα of the projectivization Pn−m−1 with the points Aα of the projective space Pn .

In our study of the structure of submanifolds in a projective space, we will often apply the method of specialization of moving frames. The idea of this method is that from all projective frames associated with an element of a submanifold, we will take the frames that are most closely connected with the element and its diﬀerential neighborhood of a certain order. Such a specialization can be conducted analytically and geometrically. Consider, for example, how the method of specialization of moving frames applies in the study of geometry of a curve on a projective plane.

41) da−1 = −a−1 da · a−1 = −ωa−1 . 40), we arrive at the equation dω = −ω ∧ ω. 43) In coordinate form, this equation is written as dωji = −ωki ∧ ωjk , or, more often, as dωji = ωjk ∧ ωki . 44) are called the structure equations or the Maurer– Cartan equations of the general linear group GL(n). 5 The Frobenius Theorem. Suppose that a system of linearly independent 1-forms θa , a = p + 1, . . , n, is given on a manifold M n . At each point x of the manifold M n , this system determines a linear subspace ∆x of the space Tx (M n ) via the equations θa (ξ) = 0.

### Differential Geometry of Varieties with Degenerate Gauss Maps by Maks A. Akivis, Vladislav V. Goldberg

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