Ivancevic_Applied-Diff-Geom

280 Applied Differential Geometry: A Modern IntroductionT m M so as to identify T m M with R n . In these coordinates one can thenexpand the metric as follows:g ij = δ ij − 1 3 R ikjlx k x l + O ( r 3) .Now the equations x i = g ij x j evidently give conditions on the curvaturesR ijkl at m.If Γ i jk(m) = 0, the manifold M is flat at the point m. This means thatthe (1, 3) curvature tensor, defined locally at m ∈ M asR l ijk = ∂ x j Γ l ik − ∂ x kΓ l ij + Γ l rjΓ r ik − Γ l rkΓ r ij,also vanishes at that point, i.e., Rijk l (m) = 0.Now, the rate of change of a vector–field A k on the manifold M alongthe curve x i (s) is properly defined by the absolute covariant derivativeDds Ak = ẋ i ∇ i A k = ẋ i ( ∂ x iA k + Γ k ij A j) = Ȧk + Γ k ij ẋ i A j .By applying this result to itself, we can get an expression for the secondcovariant derivative of the vector–field A k along the curve x i (s):D 2ds 2 Ak = d (Ȧk + Γ k ij ẋ i A j) + Γ k ij ẋ ids(Ȧj + Γ j mn ẋ m A n ).In the local coordinates (x 1 (s), ..., x n (s)) at a point m ∈ M, if δx i =δx i (s) denotes the geodesic deviation, i.e., the infinitesimal vector describingperpendicular separation between the two neighboring geodesics, passingthrough two neighboring points m, n ∈ M, then the Jacobi equation ofgeodesic deviation on the manifold M holds:D 2 δx ids 2 + R i jkl ẋ j δx k ẋ l = 0. (3.133)This equation describes the relative acceleration between two infinitesimallyclose facial geodesics, which is proportional to the facial curvature (measuredby the Riemann tensor Rjkl i at a point m ∈ M), and to the geodesicdeviation δx i . Solutions of equation (3.133) are called Jacobi fields.In particular, if the manifold M is a 2D–surface in R 3 , the Riemanncurvature tensor simplifies intoR i jmn = 1 2 R gik (g km g jn − g kn g jm ),