Then

the equality holding if and only if (8/8t)a = 0 and hence if and only if A is a geodesic curve. Thus from q to p.

The vector (d/dw)a|u_0 will be called the variation vector Z. Conversely, given a continuous, piecewise C2 vector field Z along y(t) vanishing at q and p, we may define a variation a for which Z will be the variation vector by:

a(u,t) = expr (wZ|r), where ue(-c,c) for some e > 0 and r = y(t).

Lemma 4.5.4

The variation of the length from qtop under a is BL

= ^ 't where /2 = g(S/8t, 8f8t) is the magnitude of the tangent vector and [f-18/8t] is the discontinuity at one of the singular points of y(t).

We have:

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