Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 407A variation vector–field t ↦→ W (t) is associated to each variation α,where W (t) is a tangent vector in the tangent space T γ(t) M to M, definedbyW (t) = ∂ u α(0, t). (3.225)It is a continuous map of [0, 1] into the tangent bundle T M, smooth in eachinterval [t i , t i+1 ]. These maps are the substitute for the tangent vectors atthe point γ; they form an infinite–dimensional vector space written T Ω(γ).More generally the interval ] − ε, ε[ can be replaced in the definition ofa variation by a neighborhood of 0 in some R n , defining an n−parametervariation.A critical path γ 0 ∈ Ω for a function F : Ω −→ R is defined by thecondition that for every variation α of γ 0 the functionu ↦→ F (α(u, ·))is derivable for u = 0 and its derivative is 0.Step 2 is a modern presentation of the formulas of Riemannian geometry,giving the first variation and second variation of the energy (3.224) of apath γ 0 ∈ Ω, which form the basis of Jacobi results.First consider an arbitrary path ω 0 ∈ Ω, its velocity ˙ω(t) = dω/dt, andits acceleration in the Riemannian sense¨ω(t) = ∇ t ˙ω(t),where ∇ t denotes the Bianchi covariant derivative. They belong to T ω(t) Mfor each t ∈ [0, 1], are defined and continuous in each interval [t i , t i+1 ] inwhich ω is smooth, and have limits at both extremities. Now let α be avariation of ω and t ↦→ W (t) be the corresponding variation vector–field(3.225). The first variation formula gives the first derivative1 d2 du E(α(u, ·))| u=0 = − ∑ i(W (t i )| ˙ω(t i +) − ˙ω(t i −)) −∫ 10(W (t)|¨ω(t)) dt,where (x|y) denotes the scalar product of two vectors in a tangent space.It follows from this formula that γ 0 ∈ Ω is a critical path for E iff γ is ageodesic.Next, fix such a geodesic γ and consider a two–parameter variation:α : U × [0, 1] → M,