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By Luther Pfahler Eisenhart

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Example text

Let Grn Rq0 be the Grassmanian manifold of n-dimensional linear subspaces in Rq . Let us recall that there exists a canonical Euclidean connection ∇Γ = P ◦ ∇ on the tautological n-dimensional fibration Γ → Gr n Rq0 , where E : Γ → S = (Grn Rq0 ) × Rq is the canonical embedding, P : S → E the orthogonal projection and ∇ the standard connection on S. This canonical construction is applicable also to Grn W and hence any tangential homotopy Gt can be canonically covered by an isotopy of fiberwise isometric monomorphisms Ft : T V → T W .

N, the isotopy ft satisfies the required approximation property. Let Grn Rq0 be the Grassmanian manifold of n-dimensional linear subspaces in Rq . Let us recall that there exists a canonical Euclidean connection ∇Γ = P ◦ ∇ on the tautological n-dimensional fibration Γ → Gr n Rq0 , where E : Γ → S = (Grn Rq0 ) × Rq is the canonical embedding, P : S → E the orthogonal projection and ∇ the standard connection on S. This canonical construction is applicable also to Grn W and hence any tangential homotopy Gt can be canonically covered by an isotopy of fiberwise isometric monomorphisms Ft : T V → T W .

Xn ), y = (y1 , . . , yq ), and zα = (z1,α , . . , zq,α ) are coordinates in J r (Rn , Rq ). This way any system of differential equations can be thought of as a subset of the jet space J r (Rn , Rq ). ◮ 47 48 5. Differential Relations Roughly speaking, differential equations and system of differential equations correspond to submanifolds of codimension ≥ 1 in the jet-space J 1 (Rn , Rq ), while (strict) differential inequalities correspond to open subsets. ◭ Exercise. Draw differential relations in J 1 (R, R), which correspond to the differential equation y ′ = y 2 and the differential inequality y ′ ≥ y 2 .

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