Ivancevic_Applied-Diff-Geom

20 Applied Differential Geometry: A Modern Introductionnian geometry states that there is a unique connection which satisfies theseproperties. 23In the theory of Riemannian and pseudo–Riemannian manifolds theterm covariant derivative is often used for the Levi–Civita connection. Thecoordinate expression of the connection is given by Christoffel symbols. 24Note that connection is not a tensor, except on jet bundles.The Fundamental Riemannian TensorsThe two basic objects in Riemannian geometry are the metric tensor andthe curvature tensor. The metric tensor g = 〈·, ·〉 is a symmetric second–order (i.e., (0, 2)) tensor that is used to measure distance in a space. In otherwords, given a Riemannian manifold, we make a choice of a (0, 2)−tensoron the manifold’s tangent spaces. 25 At a given point in the manifold, thistensor takes a pair of vectors in the tangent space to that point, and gives a23 The Levi–Civita connection defines also a derivative along curves, usually denotedby D. Given a smooth curve (a path) γ = γ(t) : R → M and a vector–field X = X(t)on γ, its derivative along γ is defined by: D tX = ∇ ˙γ(t) X. This equation defines theparallel transport for a vector–field X.24 The Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinateexpressions for the Levi–Civita connection derived from the metric tensor. TheChristoffel symbols are used whenever practical calculations involving geometry must beperformed, as they allow very complex calculations to be performed without confusion.In particular, if we denote the unit vectors on M as e i = ∂/∂x i , then the Christoffelsymbols of the second kind are defined by Γ k ij = 〈∇e ; e i j, e k 〉. Alternatively, using themetric tensor g ik (see below) we get the explicit expression for the Christoffel symbolsin a holonomic coordinate basis:Γ i kl = 1 2 gim „ ∂gmk∂x l+ ∂g ml∂x k− ∂g «kl∂x m .In a general, nonholonomic coordinates they include the additional commutation coefficients.The Christoffel symbols are used to define the covariant derivative of varioustensor–fields, as well as the Riemannian curvature. Also, they figure in the geodesicequation:d 2 x idt 2 + dx j dx kΓi jk = 0dt dtfor the curve x i = x i (t) on the smooth manifold M.25 The most familiar example is that of basic high–school»geometry:–the 2D Euclidean1 0metric tensor, in the usual x − y coordinates, reads: g = . The associated length0 1of a curve is given by the familiar calculus formula: L = R b pa (dx) 2 + (dy) 2 .The unit sphere in R 3 comes equipped with a natural metric induced from the ambientEuclidean»metric.–In standard spherical coordinates (θ, φ) the metric takes the form:1 0g =0 sin 2 , which is usually written as: g = dθ 2 + sinθ2 θ dφ 2 .