By Maks A. Akivis, Vladislav V. Goldberg
During this e-book the authors learn 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. via those tools, the authors notice the constitution of sorts with degenerate Gauss maps, ensure the singular issues and singular types, locate focal pictures and build a type of the kinds with degenerate Gauss maps.The authors introduce the above pointed out equipment and follow them to a sequence of concrete difficulties bobbing up within the thought of types with degenerate Gauss maps. What makes this e-book designated is the authors' use of a scientific program of tools of projective differential geometry besides equipment of the classical algebraic geometry for learning forms with degenerate Gauss maps. This e-book is meant for researchers and graduate scholars drawn to projective differential geometry and algebraic geometry and their functions. it may be used as a textual content for complicated undergraduate and graduate scholars. either authors have released over a hundred papers every 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 by means of them together.
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Additional resources for Differential Geometry of Varieties with Degenerate Gauss Maps
Conversely, if ω is a 1-form given in a simply connected domain of a manifold M n and such that dω = 0, then ω = df . 12 1. Foundational Material A p-form θ satisfying the condition dθ = 0 is called closed, and a p-form θ satisfying the condition θ = dσ, where σ is a (p − 1)-form, is called exact. , in M n every p-dimensional contour is homotopic to zero), then in M n any closed p-form is exact. This theorem follows from the p-dimensional Stokes theorem. , this operation is invariant; it commutes with the operation of coordinate transformation on the manifold M n .
M, u = 1, . . , n, between coordinates of points x ∈ Ua and y ∈ Ub . A mapping f is diﬀerentiable of class p, f ∈ C p , if and only if the functions f u are diﬀerentiable scalar 8 1. Foundational Material functions of the same class p. If the functions f u are inﬁnitely diﬀerentiable functions, then the mapping f belongs to class C ∞ , and if the functions f u are analytic functions, then f ∈ C ω . Consider the matrix ∂y u M= ∂xi having n rows and m columns. This matrix is called the Jacobi matrix of the mapping f .
143) and substituting each time the values x = 0, y (iv) y = 0, (0) = 0, y (v) y (0) = 0, = a5 , y (vi) y (0) = 1, = a6 , y (vii) y (0) = 0, = a7 , we ﬁnd the following eight relations: a000 = 0, a100 = 0, 2a110 + a200 = 0, a111 + 3a120 = 0, a220 + 2a112 = 0, a5 a200 + 30a122 = 0, a6 a200 + 12a5 a120 + 30a222 = 0, a7 a200 + 14a6 a120 + 42a5 (a112 + a220 ) = 0. 139) holds), we ﬁnd from the preceding equations that a110 = λa5 , a120 = µa5 , a200 = −2λa5 , a111 = −3µa5 , 1 2 1 2 λa , a222 = λa5 a6 − a25 µ, 5 5 15 5 1 2 = − λa7 + µa6 , 3a220 = − λa7 + 2µa6 , 7 7 3a122 = 3a112 where λ and µ are arbitrary parameters.
Differential Geometry of Varieties with Degenerate Gauss Maps by Maks A. Akivis, Vladislav V. Goldberg