Ivancevic_Applied-Diff-Geom

282 Applied Differential Geometry: A Modern Introductionbe used to characterize geodesics. Now suppose that we have a unit speedgeodesic γ (t) parameterized on [0, l] and consider a variation V (s, t) , whereV (0, t) = γ (t). Synge then shows that (¨L ≡ d2 L¨L(0) =∫ l0ds 2 ){g(Ẋ, Ẋ) − (g(Ẋ, ˙γ))2 − g(R(X, ˙γ)X, ˙γ)}dt + g( ˙γ, A)| l 0 ,where X (t) = ∂V∂s(0, t) is the variational vector–field, Ẋ = ∇ ˙γ X, andA (t) = ∇ ∂V X. In the special case where the variation fixes the endpoints,∂si.e., s → V (s, a) and s → V (s, b) are constant, the term with A in it fallsout. We can also assume that the ) variation is perpendicular to the geodesicand then drop the term g(Ẋ, ˙γ . Thus, we arrive at the following simpleform:¨L(0) =∫ l0{g(Ẋ, Ẋ) − g (R (X, ˙γ) X, ˙γ)}dt =∫ l0{|Ẋ|2 − sec( ˙γ, X) |X| 2 }dt.Therefore, if the sectional curvature is nonpositive, we immediately observethat any geodesic locally minimizes length (that is, among close–by curves),even if it does not minimize globally (for instance γ could be a closedgeodesic). On the other hand, in positive curvature we can see that if ageodesic is too long, then it cannot minimize even locally. The motivationfor this result comes from the unit sphere, where we can consider geodesicsof length > π. Globally, we know that it would be shorter to go in theopposite direction. However, if we consider a variation of γ where thevariational field looks like X = sin ( )t · πl E and E is a unit length parallelfield along γ which is also perpendicular to γ, then we get∫ l{ ∣∣∣ } ¨L(0) = Ẋ∣∣ 2 − sec ( ˙γ, X) |X| 2 dt==0∫ l0∫ l0{ (π ) ( 2· cos2t · π )ll( (π ) ( 2· cos2t · πll( − sec ( ˙γ, X) sin 2 t · π ) } dtl) ( − sin 2 t · π ) ) dt = − 1 (l 2 − π 2) ,l 2lwhich is negative if the length l of the geodesic is greater than π. Therefore,the variation gives a family of curves that are both close to and shorter thanγ. In the general case, we can then observe that if sec ≥ 1, then for thesame type of variation we get¨L(0) ≤ − 1 (l 2 − π 2) .2l