Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 313all curves on [0, 1] and proportionally to arc–length. We shall think ofΩ mp as an infinite–dimensional manifold. For each curve γ ∈ Ω mp thenatural choice for the tangent space consists of the vector–fields along γwhich vanish at the endpoints of γ. This is because these vector–fields areexactly the variational fields for curves through γ in Ω mp , i.e., fixed endpointvariations of γ. An inner product on the tangent space is then naturallydefined by(X, Y ) =∫ 10g (X, Y ) dt.Now the first variation formula for arc–length tells us that the gradient forL at γ is -∇ ˙γ ˙γ. Actually this cannot be quite right, as -∇ ˙γ ˙γ does not vanishat the endpoints. The real gradient is gotten in the same way we find thegradient for a function on a surface in space, namely, by projecting it downinto the correct tangent space. In any case we note that the critical pointsfor L are exactly the geodesics from m to p. The second variation formulatells us that the Hessian of L at these critical points is given by∇ 2 L (X) = Ẍ + R (X, ˙γ) ˙γ,at least for vector–fields X which are perpendicular to γ. Again we ignorethe fact that we have the same trouble with endpoint conditions as above.We now need to impose the Morse condition that this Hessian is not allowedto have any kernel. The vector–fields J for which ¨J + R (J, ˙γ) ˙γ = 0 arecalled Jacobi fields. Thus we have to Figure out whether there are anyJacobi fields which vanish at the endpoints of γ. The first observation isthat Jacobi fields must always come from geodesic variations. The Jacobifields which vanish at m can therefore be found using the exponential mapexp m . If the Jacobi field also has to vanish at p, then p must be a criticalvalue for exp m . Now Sard’s Theorem asserts that the set of critical valueshas measure zero. For given m ∈ M it will therefore be true that thearc–length functional on Ω mp is a Morse function for almost all p ∈ M.Note that it may not be possible to choose p = m, the simplest examplebeing the standard sphere. We are now left with trying to decide what theindex should be. This is the dimension of the largest subspace on whichthe Hessian is negative definite. It turns out that this index can also becomputed using Jacobi fields and is in fact always finite. Thus one cancalculate the topology of Ω mp , and hence M, by finding all the geodesicsfrom m to p and then computing their index.