# The Poincare Half-Plane: A Gateway to Modern Geometry by Saul Stahl

By Saul Stahl

Stahl's moment version maintains to supply scholars with the uncomplicated and optimistic improvement of contemporary geometry that brings them towards present geometric study. even as, repeated use is made up of highschool geometry, algebra, trigonometry, and calculus, hence reinforcing the scholars' realizing of those disciplines in addition to improving their conception of arithmetic as a unified activity. This exact method makes those complicated geometry rules obtainable to undergraduates and graduates alike.

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N) , more precisely, n ai (x)Γ(i) (x) = 0. Γ(n+1) (x) + i=0 This already gives us a differential operator depending, however, on the lift. Let us find a new, canonical, lift for which the Wronski determinant identically equals 1. Any lift of γ(x) is of the form α(x)Γ(x) for some nonvanishing function α(x). The condition on this function is |αΓ, (αΓ) , . . , (αΓ)(n) | = 1, and hence α(x) = W (x)−1/(n+1) . 1). This operator is uniquely defined by the curve γ(x). 1. 1) if and only if they are projectively equivalent.

4) is a projective differential invariant of the curve. 4, is actually invariant under the full group Diff(S 1 ). The constructed set of projective invariants is complete. Indeed, the differential operator A characterizes the curve up to projective equivalence, while the total symbol completely determines the operator. Comment The theory of projective differential invariants of non-degenerate curves is a classical subject. It was thoroughly studied in XIX-th century and summarized as early as in 1906 by Wilczynski [231].

A) Equivariance with respect to the vector field d/dx implies that all the coefficients C j,m (x) in σλ,µ are constants. 3). (c) Consider the third vector field x 2 d/dx. 2. Check by a straightforward computation that equivariance with respect to the vector field x2 d/dx leads to the recurrence relation C j+1 +1 = (j + 1)(j + 2λ) Cj. 4). 2) guarantees that the denominator in the recurrence formula does not vanish. 4) can by written in a more elegant way. Let us use the following standard notation: a differential operator A = j aj (x)(d/dx)j is denoted by j aj (x)ξ j , where ξ is a formal variable.