Bourbaki

Book VAR · English (translated)

Index of statements

The 6 statements of Differentiable and Analytic Manifolds, by chapter and §. Pick a kind, filter by a word, or put the most cited first.

2. Variétés différentielles et analytiques, fascicule de résultats

§ 9. Differential equations and foliations

  • Example Let K = R and X = R². Let the discrete group Γ = Z² operate on X by translations. Let m ∈ R and let (X, Yₘ) be the foliation defined by the submersion p : (x₁, x₂) ↦ x₂ - mx₁ of X… 0EEH
  • Example If F has rank 1 at every point, F is integrable, and defines on X a pure foliation of dimension 1. Suppose in addition that K = R, that X is separated, and that F has as a frame a… 0EEI

§ 11. Stokes Formula

  • Example When X = Rⁿ, ξ is the usual orientation of Rⁿ and A is a closed ball of radius > 0, the orientation of the sphere ∂ A defined by ξ is the canonical orientation (10.2.8, b)). 0EEJ

§ 12. Jets

  • Example Take X = Kⁿ, and denote by u₁, …, uₙ the coordinate functions on Kⁿ; then the fibre Pᵏ₀(X) of Pᵏ(X) at 0 has as basis the family of jets of order k of the monomials u₁^m₁ … uₙ^mₙ… 0EEK

§ 14. Differential Operators

  • Example Let J be a finite set, and X an open set of Kᴶ. Let H be a scalar n-differential operator of order ≤ k on X. There exists a unique family of scalar functions c(α(1), …, α(n))_(α(1)… 0EEL
  • Example Let X be an open subset of Kⁿ, let ξ¹,…,ξⁿ denote the coordinate functions on X, and identify Ω with K_X by means of the frame ω=dξ¹∧⋯∧ dξⁿ. If E=F=K_X, one has 0EEM