Ivancevic_Applied-Diff-Geom

410 Applied Differential Geometry: A Modern Introductiondone by considering the geodesic arc γ τ : [0, τ] → M, the restriction of γto [0, τ], and its energyE(γ τ ) = τ∫ τ0‖ dγdu ‖2 du.E∗∗ τ is the corresponding quadratic form on T Ω(γ τ ), and λ(τ) is its index;one studies the variation of λ(τ) when tau varies from 0 to 1, and λ(1) isthe index of E ∗∗ .The index Theorem says: the index of E ∗∗ is the sum of the multiplicitiesof the points conjugate to A along B and distinct from B.We have seen that the dimension of T Ω(γ; t 0 , t 1 , · · · , t m ) is finite; itfollows that the index of E ∗∗ is always finite, and therefore the number ofpoints conjugate to A along γ is also finite.Step 4 of Morse theory introduces a topology on the set Ω = Ω(M; A, B).On the biodynamical configuration manifold M the usual topology can bedefined by a distance ρ(A, B), the g.l.b. of the lengths of all piecewisesmooth paths joining A and B. For any pair of paths ω 1 , ω 2 in Ω(M; A, B),consider the function d(ω 1 , ω 2 ) ∈ M√ ∫ 1d(ω 1 , ω 2 ) = sup ρ(ω 1 (t), ω 2 (t)) + (ṡ 1 − ṡ 2 ) 2 dt,0t1where s 1 (t) (resp. s 2 (t)) is the length of the path τ ↦→ ω 1 (τ) (resp. τ ↦→ω 2 (τ)) defined in [0, t]. This distance on Ω such that the function ω ↦→EA B (ω) is continuous for that distance.Morse Homology of a Biodynamical ManifoldMorse Functions and Boundary Operators. Let f : M → Rrepresents a C ∞ −function on the biodynamical configuration manifold M.Recall that z = (q, p) ∈ M is the critical point of f if df(z) ≡ df[(q, p)] = 0.In local coordinates (x 1 , ..., x n ) = (q 1 , ..., q n , p 1 , ..., p n ) in a neighborhoodof z, this means ∂f∂x(z) = 0 for i = 1, ..., n. The Hessian of f at a criticalipoint z defines a symmetric bilinear form ∇df(z) = d 2 f(z) ( on T z M, inlocal coordinates (x 1 , ..., x n ) represented by the matrix ∂ 2 f∂x i ∂x). Indexjand nullity of this matrix are called index and nullity of the critical pointz of f.Now, we assume that all critical points z 1 , ..., z n of f ∈ M are nondegeneratein the sense that the Hessians d 2 f(z i ), i = 1, ..., m, have maximalrank. Let z be such a critical point of f of Morse index s (= number0