Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 143on the subject.M ... the space–time manifold Mg ij = g ij (x i ) ∈ T x M ... metric tensor on Mg ij = (g ij ) −1 ... inverse metric tensor on MΓ ijk = 1 2 (∂ x kg ij + ∂ x j g ki − ∂ x ig jk )... 1–order Christoffel symbolsΓ k ij = g kl Γ ijl ... 2–order Christoffel symbols (Levi–Civita connection)Rijk l = ∂ x j Γ l ik − ∂ x kΓ l ij + Γ l rjΓ r ik − Γ l rkΓ r ij ... Riemann curvature tensorR ij = Rijll ... Ricci tensor is the trace of RiemannR = g ij R ij ... scalar curvature is the trace of RicciG ij = R ij − 1 2 Rg ij... Einstein tensor is the trace–reversed RicciT ij = −2 δL Hilbδg ij + g ij L Hilb ... stress–energy–momentum (SEM) tensorL Hilb = 116π gij R ij (−g) 1/2∫... is derived from the Hilbert LagrangianδS = δ L Hilb (−g) 1/2 d 4 x = 0 ... the Hilbert action principle givesG ij = 8πT ij... the Einstein equation.We will continue Einstein’s geometrodynamics in section 6.4 below.3.2 Intuition Behind the Manifold ConceptAs we have already got the initial feeling, in the heart of applied differentialgeometry is the concept of a manifold. As a warm–up, to get some dynamicalintuition behind this concept, let us consider a simple 3DOF mechanicalsystem determined by three generalized coordinates, q i = {q 1 , q 2 , q 3 }.There is a unique way to represent this system as a 3D manifold, such thatto each point of the manifold there corresponds a definite configuration ofthe mechanical system with coordinates q i ; therefore, we have a geometricalrepresentation of the configurations of our mechanical system, calledthe configuration manifold. If the mechanical system moves in any way,its coordinates are given as the functions of the time. Thus, the motionis given by equations of the form: q i = q i (t). As t varies (i.e., t ∈ R), weobserve that the system’s representative point in the configuration manifolddescribes a curve and q i = q i (t) are the equations of this curve.