Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 273More precisely, a vector–field X along a parameterized curve α : I → Min M is tangent to M along α if X(t) ∈ M α(t) for all for t ∈ I ⊂ R. However,the derivative Ẋ of such a vector–field is, in general, not tangent to M.We can, nevertheless, get a vector–field tangent to M by projecting Ẋ(t)orthogonally onto M α(t) for each t ∈ I. This process of differentiating andthen projecting onto the tangent space to M defines an operation with thesame properties as differentiation, except that now differentiation of vector–fields tangent to M induces vector–fields tangent to M. This operation iscalled covariant differentiation.Let γ : I → M be a parameterized curve in M, and let X be a smoothvector–field tangent to M along α. The absolute covariant derivative ofX is the vector–field ˙¯X tangent to M along α, defined by ˙¯X = Ẋ(t) −[Ẋ(t) · N(α(t))] N(α(t)), where N is an orientation on M. Note that ˙¯X isindependent of the choice of N since replacing N by -N has no effect onthe above formula.Lie bracket (3.7.2) defines a symmetric affine connection ∇ on any manifoldM:[X, Y ] = ∇ X Y − ∇ Y X.In case of a Riemannian manifold M, the connection ∇ is also compatiblewith the Riemannian metrics g on M and is called the Levi–Civitaconnection on T M.For a function f ∈ C k (M, R) and a vector a vector–field X ∈ X k (M)we always have the Lie derivative (3.7)L X f = ∇ X f = df(X).But there is no natural definition for ∇ X Y, where Y ∈ X k (M), unlessone also has a Riemannian metric. Given the tangent field ˙γ, the accelerationcan then be computed by using a Leibniz rule on the r.h.s, if we canmake sense of the derivative of ∂ x i in the direction of ˙γ. This is exactlywhat the covariant derivative ∇ X Y does. If Y ∈ T m M then we can writeY = a i ∂ x i, and therefore∇ X Y = L X a i ∂ x i. (3.127)Since there are several ways of choosing these coordinates, one must checkthat the definition does not depend on the choice. Note that for two vector–fields we define (∇ Y X)(m) = ∇ Y (m) X. In the end we get a connection∇ : X k (M) × X k (M) → X k (M),