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Pick tn ↓ α as n ↑ ∞. Then, (tn ) is a Cauchy sequence and for n, m we have d(γξ (tn ), γξ (tm )) ≤ (γξ |[tn , tm ]) = |ξ ||tn − tm |; thus, γξ (tn ) is also a Cauchy sequence in M and γξ (tn ) → q for some q ∈ M, as n → ∞. But, for a relatively compact neighborhood U = U (q) of q, we have {ζ ∈ π −1 [U ] : |ζ | = |ξ |} is compact in T M. 8. Calculations with Moving Frames 29 (tn ), and η ∈ Mq for which γξ (τk ) → η as k → ∞. If we set γη (t) = exp tη, then for all k we have γη (τk − α) = γξ (τk ) by the uniqueness of integral curves of the geodesic flow.

A more general result is: Theorem (See Rinow (1961, p. 9. Notes and Exercises 41 (note that, by Rademacher’s theorem – see Chavel (2001, p. 20ff) – f exists for almost all t in [a, b]). 7). 11. The method of moving frames goes back at least to G. Darboux’s (1898) triply orthogonal systems, was turned into a modern tool in differential ´ Cartan (1946), and widely disseminated in the geometry in the works of E. S. Chern. An attractive entr´ee into the method can be found in Flanders (1963). Hessians Definition.

The case of equality is handled easily. 2. Pick 0 > 0 sufficiently small so that ( 0 + 1) (γ ) < ; then, γ can be extended to a geodesic γ : [− 0 , 1] → U . 1 with respect to γ (− 0 ). Let G be the image under exp of the open annulus in Mγ (− 0 ) given by B(γ (− 0 ); (γ ))\B(γ (− 0 ); 0 ), that is, G = exp B(γ (− 0 ); (γ ))\B(γ (− 0 ); 0 ). Then, {τ : ω(τ ) ∈ G} is open in R and is given by a countable disjoint union of open intervals {(α j , β j ) : j = 1, . }. 2 to each [α j , β j ], we have (ω) ≥ (ω|[α j , β j ]) ≥ j |t j (β j ) − t j (α j )|, j where t j is the function t(τ ), above, for each ω|[α j , β j ].

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