Ivancevic_Applied-Diff-Geom

278 Applied Differential Geometry: A Modern Introductionfixes the endpoints, X (a) = X (b) = 0, then the second term in the formuladrops out, and we note that the length of γ can always be decreased as longas the acceleration of γ is not everywhere zero. Thus the Euler–Lagrangianequations for the arc–length functional are the equations for a curve to bea geodesic.Recall that in local coordinates x i ∈ U, where U is an open subset in theRiemannian manifold M, the geodesics are defined by the geodesic equationẍ i + Γ i jkẋ j ẋ k = 0, (3.132)where overdot means derivative upon the line parameter s, while Γ i jk areChristoffel symbols of the affine Levi–Civita connection ∇ on M. From(3.132) it follows that the linear connection homotopy,¯Γ i jk = sΓ i jk + (1 − s)Γ i jk, (0 ≤ s ≤ 1),determines the same geodesics as the original Γ i jk .3.10.1.3 Riemannian Curvature on MThe Riemann curvature tensor is a rather ominous tensor of type (1, 3);i.e., it has three vector variables and its value is a vector as well. It isdefined through the Lie bracket (3.7.2) asR (X, Y ) Z = ( ∇ [X,Y ] − [∇ X , ∇ Y ] ) Z = ∇ [X,Y ] Z − ∇ X ∇ Y Z + ∇ Y ∇ X Z.This turns out to be a vector valued (1, 3)−tensor–field in the three variablesX, Y, Z ∈ X k (M). We can then create a (0, 4)−tensor,R (X, Y, Z, W ) = g ( ∇ [X,Y ] Z − ∇ X ∇ Y Z + ∇ Y ∇ X Z, W ) .Clearly this tensor is skew–symmetric in X and Y , and also in Z andW ∈ X k (M). This was already known to Riemann, but there are somefurther, more subtle properties that were discovered a little later by Bianchi.The Bianchi symmetry condition readsR(X, Y, Z, W ) = R(Z, W, X, Y ).Thus the Riemann curvature tensor is a symmetric curvature operatorR : Λ 2 T M → Λ 2 T M.The Ricci tensor is the (1, 1)− or (0, 2)−tensor defined byRic(X) = R(∂ x i, X)∂ x i, Ric(X, Y ) = g(R(∂ x i, X)∂ x i, Y ),