By Detlef Laugwitz (Auth.)

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For every point x(u ) of a surface and for every direction x on the surface through that point, there is one and only one geodesic through the given point in the given direction. Proof. Let the direction on the surface be given by its unit vector ο = v x with υ = g (u )v v = 1. 16) of differential equations, 2 % it % % k ik «*"= - rj (u')ui'u ' k k with the initial conditions w*(0) = u\ u (0) = v\ % From the existence and uniqueness theorem for systems of ordinary differential equations, it follows that there exists one and only one solution u\s) satisfying the initial conditions.

Which leave x, xv x2 invariant, thus t h a t F cannot explicitly depend on the w*. translation in space, χ varies but the Thus F = F(x, xv x2). do not ; hence F = F(xv x2). Now recognizing under a Finally, be means of a suitable parameter transformation the Xi can be transformed into any other pair of vectors. 5 The Curvature of Curves on the Surface Having made use of arc length, we shall next exploit the curvature of curves on the surface in a similar way for studying the geometry of the surface.

Then the angle between v and w is constant along the curve, and the vectors v (t) are all of equal length. x x x f % % 50 2. L O C A L D I F F E R E N T I A L G E O M E T R Y O F S U R F A C E S Proof. 4). This proves first that lengths remain constant (apply the last equation to the special case v = w ) and then that angles remain constant, as was to be proved. % % F I G . 18. Parallel vector field on the sphere. We re-emphasize that geodesic parallelism of vectors is only defined with respect to (or along) a curve.