Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 415Following the standard algebraic construction, we define (relative)Morse cohomology. We setand defineC p λ (f) = Hom(Cλ p (f), Z),δ λand: C p λ(f) → Cp+1λ(f), 〈δ λ a, x〉 = 〈a, ∂ λ x〉H p λ (f) = Ker(δλ )/Im(δ λ ).Since Crit p (f) is finite, we have H λ p (f) = H p (f) and H p λ (f) = Hp (f).3.13.5.3 Hodge–De Rham Theory in BiodynamicsHodge LaplacianA single biodynamical configuration manifold M can be equipped withmany different Riemannian metrics g in local coordinates (apart from theone generated by its kinetic energy)g = g ij (u 1 , u 2 , ..., u n ) du i du j .Beltrami had shown that it is always possible for such a metric to definean operator (depending on the metric) that generalizes the usual Laplacianon R n and therefore induces the notion of harmonic functions on theRiemannian manifold [Choquet-Bruhat and DeWitt-Morete (1982)].Hodge theory was described by H. Weyl as ‘one of the landmarks inthe history of mathematics in the 20th Century’. Hodge showed that it waspossible to define a notion of harmonic exterior differential form: the metricg on M canonically defines a metric on the tangent bundle T M, hence also,by standard multilinear algebra, a metric on any bundle of tensors on M. Inparticular, let (α, β) ↦→ g p (α, β) be the positive nondegenerate symmetricbilinear form defined on the vector space of p−forms on M. As M isorientable, this defines a duality between p−forms and (n − p)−forms: toeach p−form α is associated a (n−p)−form ∗α, defined by the linear Hodgestar operator ∗ (see subsection 3.6.3.7), characterized by the relationsβ ∧ (∗α) = g p (α, β) v, ∗ ∗ α = (−1) p(n−p) α,for all p−forms α, β, where v is the volume form on the Riemannian manifoldM. If d is the exterior derivative, it has a transposed (adjoint) operator