By Kapovich M.
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Extra info for 3-manifold Groups and Nonpositive Curvature
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 .
In this study, we will use both geometric and analytic variations of this method. Let Γ be a smooth simple connected curve in the projective plane P2 . A moving frame in P2 consists of three points A0 , A1 , and A2 that do not belong to a straight line. 94) (cf. 73)). 95) (cf. 70)). 96) (cf. 72)). First, we apply two geometric specializations of the moving frame. Suppose that A0 = x ∈ Γ and locate the point A1 on the tangent Tx (Γ) to Γ at the point x, A1 ∈ Tx (Γ). Then we have dA0 = ω00 A0 + ω01 A1 .
Xn0 (xui ) = x01 x11 . . xn1 . 161) .................... x0m x1m . . im of order m + 1 of this matrix: xi00 xi01 . . im = det xi10 xi11 . . xi1m . xim0 xim1 . . 162) Because the matrix has m + 1 rows and n + 1 columns, the total number of n+1 such minors is equal to m+1 . 161) also changes, but all of its minors are multiplied by the same factor, namely, the determinant of the matrix of basis transformation. Thus, these minors can be taken as homogeneous projective coordinates of a point in n+1 the projective space PN of dimension N = m+1 − 1.