Ivancevic_Applied-Diff-Geom

274 Applied Differential Geometry: A Modern Introductionwhich satisfies (for all f ∈ C k (M, R) and X, Y, Z ∈ X k (M)):(1) Y → ∇ Y X is tensorial, i.e., linear and ∇ fY X = f∇ Y X.(2) X → ∇ Y X is linear.(3) ∇ X (fY ) = (∇ X f)Y (m) + f(m)∇ X Y .(4) ∇ X Y − ∇ Y X = [X, Y ].(5) L X g(Z, Y ) = g(∇ X Z, Y ) + g(Z, ∇ X Y ).A semicolon is commonly used to denote covariant differentiation withrespect to a natural basis vector. If X = ∂ x i, then the components of ∇ X Yin (3.127) are denotedY k; i = ∂ x iY k + Γ k ij Y j , (3.128)where Γ k ij are Christoffel symbols defined in (3.129) below. Similar relationshold for higher–order tensor–fields (with as many terms with Christoffelsymbols as is the tensor valence).Therefore, no matter which coordinates we use, we can now define theacceleration of a curve in the following way:γ(t) = (γ 1 (t), . . . , γ n (t)),˙γ(t) = ˙γ i (t)∂ x i,¨γ(t) = ¨γ i (t)∂ x i + ˙γ i (t)∇ ˙γ(t) ∂ x i.We call γ a geodesic if γ(t) = 0. This is a second–order nonlinear ODE ina fixed coordinate system (x 1 , . . . , x n ) at the specified point m ∈ M. Thuswe see that given any tangent vector X ∈ T m M, there is a unique geodesicγ X (t) with ˙γ X (0) = X. If the manifold M is closed, the geodesic must existfor all time, but in case the manifold M is open this might not be so. Tosee this, take as M any open subset of Euclidean space with the inducedmetric.Given an arbitrary vector–field Y (t) along γ, i.e., Y (t) ∈ T γ(t) M for allt, we can also define the derivative Ẏ ≡ dYdtin the direction of ˙γ by writingY (t) = a i (t)∂ x i,Ẏ (t) = ȧ i (t)∂ x i + a i (t)∇ ˙γ(t) ∂ x i.Here the derivative of the tangent field ˙γ is the acceleration γ. The field Yis said to be parallel iff Ẏ = 0. The equation for a field to be parallel is afirst–order linear ODE, so we see that for any X ∈ T γ(t0)M there is a uniqueparallel field Y (t) defined on the entire domain of γ with the property that