Ivancevic_Applied-Diff-Geom

418 Applied Differential Geometry: A Modern IntroductionHodge–Weyl Theorem [Griffiths (1983b); Voisin (2002)] states that everyde Rham cohomology class has a unique harmonic representative.Heat Kernel and Thermodynamics on MBesides pure mechanical consideration of biodynamical system, there isanother biophysical point of view – thermodynamical, compatible with thehuman motion [Hill (1938)]. Namely, the heat equation on the biodynamicalconfiguration manifold M,∂ t a(t) = ∆a(t), with initial condition a(0) = α,has a unique solution for every t ∈ [0, ∞) and every p−form α on M.If we think of α as an initial temperature distribution on M then as theconfiguration manifold cools down, according to the classical heat equation,the temperature should approach a steady state which should be harmonic[Davies (1989)].To prove this, we define a stationary and hence harmonic operatorH(α) = lim t→∞ a(t). Also, a map α → G(α) withG(α) =∫ ∞0a(t) dtis orthogonal to the space of harmonic forms and satisfies∆G(α) =∫ ∞0∆a(t) dt = −∫ ∞0∂ t a(t) dt = α − H(α).Here, the map α → H(α) is called harmonic projection and the map α →G(α) is called Green’s operator.In particular, for each p−form α we get a unique decompositionα = H(α) + ∆G(α).This proves the existence of a harmonic representative in every de Rhamcohomology class, as follows.Let α ∈ Ω p (M) be a closed form. Thenα = H(α) + dd ∗ G(α) + d ∗ dG(α).But the three terms in this sum are orthogonal and so‖d ∗ dG(α)‖ = 〈d ∗ dG(α), α〉 = 〈dG(α), dα〉 = 0,since α is closed. Thus H(α) is cohomologous to α.