Ivancevic_Applied-Diff-Geom

272 Applied Differential Geometry: A Modern Introduction2−plane in a tangent space a real number. Given a line L in a tangent space,we can average the sectional curvatures of all planes through L to get theRicci tensor Rc(L). Likewise, given a point x ∈ M, we can average the Riccicurvatures of all lines in the tangent space of x to get the scalar curvatureR(x). In local coordinates, the Ricci tensor is given by R ik = g jl R ijkl andthe scalar curvature is given by R = g ik R ik , where (g ij ) = (g ij ) −1 is theinverse of the metric tensor (g ij ).3.10.1.1 Riemannian Metric on MIn this section we mainly follow [Petersen (1999); Petersen (1998)].Riemann in 1854 observed that around each point m ∈ M one can picka special coordinate system (x 1 , . . . , x n ) such that there is a symmetric(0, 2)−tensor–field g ij (m) called the metric tensor defined asg ij (m) = g(∂ x i, ∂ x j ) = δ ij , ∂ x kg ij (m) = 0.Thus the metric, at the specified point m ∈ M, in the coordinates(x 1 , . . . , x n ) looks like the Euclidean metric on R n . We emphasize thatthese conditions only hold at the specified point m ∈ M. When passing todifferent points it is necessary to pick different coordinates. If a curve γpasses through m, say, γ(0) = m, then the acceleration at 0 is defined byfirstly, writing the curve out in our special coordinatesγ(t) = (γ 1 (t), . . . , γ n (t)),secondly, defining the tangent, velocity vector–field, as˙γ = ˙γ i (t) · ∂ x i,and finally, the acceleration vector–field as¨γ(0) = ¨γ i (0) · ∂ x i.Here, the background idea is that we have a connection.Recall that a connection on a smooth manifold M tells us how to paralleltransport a vector at a point x ∈ M to a vector at a point x ′ ∈ M alonga curve γ ∈ M. Roughly, to parallel transport vectors along curves, it isenough if we can define parallel transport under an infinitesimal displacement:given a vector X at x, we would like to define its parallel transportedversion ˜X after an infinitesimal displacement by ɛv, where v is a tangentvector to M at x.