Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 281where R denotes the scalar Gaussian curvature. Consequently the equationof geodesic deviation (3.133) also simplifies intoD 2ds 2 δxi + R 2 δxi − R 2 ẋi (g jk ẋ j δx k ) = 0. (3.134)This simplifies even more if we work in a locally Cartesian coordinatesystem; in this case the covariant derivative D2Dsreduces to an ordinary2derivative d2dsand the metric tensor g 2ij reduces to identity matrix I ij , soour 2D equation of geodesic deviation (3.134) reduces into a simple second–order ODE in just two coordinates x i (i = 1, 2)ẍ i + R 2 δxi − R 2 ẋi (I jk ẋ j δx k ) = 0.3.10.2 Global Riemannian Geometry3.10.2.1 The Second Variation FormulaCartan also establishes another important property of manifolds with nonpositivecurvature. First he observes that all spaces of constant zero curvaturehave torsion–free fundamental groups. This is because any isometryof finite order on Euclidean space must have a fixed point (the center ofmass of any orbit is necessarily a fixed point). Then he notices that onecan geometrically describe the L ∞ center of mass of finitely many points{m 1 , . . . , m k } in Euclidean space as the unique minimum for the strictlyconvex function1{x → max (d (m i , x)) 2} .i=1,··· ,k 2In other words, the center of mass is the center of the ball of smallest radiuscontaining {m 1 , . . . , m k } . Now Cartan’s observation from above was thatthe exponential map is expanding and globally distance nondecreasing as amap:(T m M, Euclidean metric) → (T m M, with pull–back metric) .Thus distance functions are convex in nonpositive curvature as well as inEuclidean space. Hence the above argument can in fact be used to concludethat any Riemannian manifold of nonpositive curvature must also havetorsion free fundamental group.Now, let us set up the second variation formula and explain how itis used. We have already seen the first variation formula and how it can