By Michal A.D.

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Let u = u (u ), a ^ u ^ b, give a piecewise smooth curve in G which connects the end points of the given geodesic arc, with u (a) = u (b) = 0. 20), the length of this curve satisfies 2 2 2 x 1 2 b j V l + g 2 "id" 1 2 a which proves the theorem in the restricted sense that only a restricted class of curves has so far been admitted for comparison with the given geodesic arc. To complete the proof, we must also admit curves that may not be representable by an equation u = w (w ) and which may, therefore, intersect a curve u = const, more than once.

16). Toward this, in turn, we consider the function f(s) = &*(«*(s))«*V and we will show that /'(s) = 0. As the initial condition g {u )v v = /(0) = 1 % j % k ik then yields f(s) = 1, we will have shown that s is the arc length. Now 4. I N T R I N S I C G E O M E T R Y O F f'(s) = -ß- ui' u u»' + if gik 43 SURFACES u>" «*' + u «*". 16); therefore, f'(s) = 0. 4 The Extremal Property of the Geodesic Curves A straight line of the plane is characterized by its property of being the shortest possible connecting curve between any two of its points.

They are expressible in terms of the functions g alone, as follows. We have g = x x . Differentiating with respect to u , we obtain k i k i k { k l -J7- = *•/ *Jk + **/ *i = R*u\K + Ai|». 4) 4. I N T R I N S I C G E O M E T R Y 39 OF SURFACES We note down also the corresponding equations obtained by permuting the indices, = s** */ + si* *. 4 ) *tfXft, and we obtain from these, on account of the equality of certain pairs of second mixed partial derivatives, that ι /Hn Hik J _ Hik\ „ ρ x Let us denote by (g ) the matrix inverse to (g ); it exists, since tk iÄ = g > 0.