Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 519(1985)]).A differential operator of order m, mapping the smooth sections of avector bundle E over a compact manifold Y to those of another such bundleF , can be described in local coordinates and local trivializations of thebundles asD = ∑a α (x)D α ,|α|≤mwith α = (α 1 , . . . , α n ). The coefficients a α (x) are matrices of smoothfunctions that represent elements of Hom(E, F ) locally; and D α =∂ ∂∂x α 1 · · ·∂x . αn1nThe principal symbol associated to the operator D is the expressionσ m (D)(x, p) = ∑a α (x)p α .|α|=mGiven the differential operator D : Γ(Y, E) → Γ(Y, F ), the principalsymbol with the local expression above defines a global map σ m : π ∗ (E) →π ∗ (F ), where T ∗ Y π → Y is the cotangent bundle; that is, the variables (x, p)are local coordinates on T ∗ Y .Consider bundles E i , i = 1 . . . k, over a compact nD manifold Y suchthat there is a complex Γ(Y, E) formed by the spaces of sections Γ(Y, E i )and differential operators d i of order m:0 → Γ(Y, E 1 ) → d1· · · dk−1→ Γ(Y, E k ) → 0.Construct the principal symbols σ m (d i ); these determine an associatedsymbol complex0 → π ∗ (E 1 ) σm(d1)→· · · σm(d k−1)→ π ∗ (E k ) → 0.The complex Γ(Y, E) is elliptic iff the associated symbol complex isexact.differential equations, involving their rates of change. Such formulas may havean ‘index’, the number of solutions of the formulas minus the number of restrictionsthat they impose on the values of the quantities being computed. TheAtiyah–Singer index Theorem calculated this number in terms of the geometryof the surrounding space. A simple case is illustrated by a famous paradoxicaletching of M. C. Escher, ‘Ascending and Descending’, where the people, goinguphill all the time, still manage to circle the castle courtyard. The indexTheorem would have told them this was impossible.”