Ivancevic_Applied-Diff-Geom

276 Applied Differential Geometry: A Modern IntroductionFrom the skew symmetry ([X, Y ] = −[Y, X]) of the Lie bracket, follows theskew symmetry (T (X, Y ) = −T (Y, X)) of the torsion tensor. The mappingT is said to be f−bilinear since it is linear in both arguments and alsosatisfies T (fX, Y ) = fT (X, Y ) for smooth functions f. Since [∂ x i, ∂ x j ] = 0for all 1 ≤ i, j ≤ n, it follows thatT (∂ x i, ∂ x j ) = (Γ k ij − Γ k ji)∂ x k.Consequently, torsion T is a (1, 2) tensor–field, locally given byT = T k i j dx i ⊗ ∂ x k ⊗ dx j ,where the torsion components T k i jare given byT k i j = Γ k ij − Γ k ji.Therefore, the torsion tensor gives a measure of the nonsymmetry of theconnection coefficients. Hence, T = 0 if and only if these coefficients aresymmetric in their subscripts. A connection ∇ with T = 0 is said to betorsion free or symmetric.The connection also enables us to define many other classical conceptsfrom calculus in the setting of Riemannian manifolds. Suppose we have afunction f ∈ C k (M, R). If the manifold is not equipped with a Riemannianmetric, then we have the differential of f defined by df(X) = L X f, whichis a 1−form. The dual concept, the gradient of f, is supposed to be avector–field. But we need a metric g to define it. Namely, ∇f is defined bythe relationshipg(∇f, X) = df(X).Having defined the gradient of a function on a Riemannian manifold, wecan then use the connection to define the Hessian as the linear map∇ 2 f : T M → T M,∇ 2 f(X) = ∇ X ∇f.The corresponding bilinear map is then defined as∇ 2 f(X, Y ) = g(∇ 2 f(X), Y ).One can check that this is a symmetric bilinear form. The Laplacian of f,∆f, is now defined as the trace of the Hessian∆f = Tr(∇ 2 f(X)) = Tr(∇ X ∇f),