Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Mathematics, Frobenius' theorem gives Necessary and sufficient conditions for finding a maximal set of independent solutions of an Overdetermined system It is a foundational result in several fields, the chief among them being symplectic geometry. Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a The theorem is named after Jean Gaston Darboux^[1] who established it as the solution of the Pfaff problem. Jean-Gaston Darboux ( August 14, 1842, Nîmes  &ndash February 23, 1917, Paris) was a French Mathematician Johann Friedrich Pfaff (sometimes spelled Friederich) was born in Stuttgart on December 22, 1765, and died in Halle on April ^[2]

One of the many consequences of the theorem is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In Mathematics, a symplectomorphism is an Isomorphism in the category of Symplectic manifolds Formal definition Specifically That is, every 2n-dimensional symplectic manifold can be made to look locally like the linear symplectic space Cⁿ with its canonical symplectic form. In Mathematics, a symplectic vector space is a Vector space V equipped with a Nondegenerate, Skew-symmetric, Bilinear form There is also an analogous consequence of the theorem as applied to contact geometry. In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle

Statement and first consequences

The precise statement is as follows. ^[3] Suppose that θ is a differential 1-form on an n dimensional manifold, such that dθ has constant rank p. If

θ ∧ (dθ)^p = 0 everywhere,

then there is a local system of coordinates x₁,. . . ,x_n-p, y₁, . . . , y_p in which

θ = x₁ dy₁ + . . . + x_p dy_p.

If, on the other hand,

θ ∧ (dθ)^p ≠ 0 everywhere,

then there is a local system of coordinates x₁,. . . ,x_n-p, y₁, . . . , y_p in which

θ = x₁ dy₁ + . . . + x_p dy_p + dx_p+1.

In particular, suppose that ω is a symplectic 2-form on an n=2m dimensional manifold M. In a neighborhood of each point p of M, by the Poincaré lemma, there is a 1-form θ with dθ=ω. In Mathematics, especially Vector calculus and Differential topology, a closed form is a Differential form α whose differential is Moreover, θ satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart U near p in which

θ = x₁ dy₁ + . For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how . . + x_m dy_m.

Taking an exterior derivative now shows

ω = dθ = dx₁ ∧ dy₁ + . In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms . . + dx_m ∧ dy_m.

The chart U is said to be a Darboux chart around p. ^[4] The manifold M can be covered by such charts. In Mathematics, a cover of a set X is a collection of sets such that X is a Subset of the union of sets in the collection

To state this differently, identify R^2m with C^m by letting z_j = x_j + i y_j. If φ : U → Cⁿ is a Darboux chart, then ω is the pullback of the standard symplectic form ω₀ on Cⁿ:

. Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from

Comparison with Riemannian geometry

This result implies that there are no local invariants in symplectic geometry: a Darboux basis can always be taken, valid near any given point. This is in marked contrast to the situation in Riemannian geometry where the curvature is a local invariant, an obstruction to the metric being locally a sum of squares. Elliptic geometry is also sometimes called Riemannian geometry. In Mathematics, specifically Differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space

It should be emphasized that the difference is that Darboux's theorem states that ω can be made to take the standard form in an entire neighborhood around p. In Riemannian geometry, the metric can always be made to take the standard form at any given point, but not always in a neighborhood around that point.

See also

• Carathéodory-Jacobi-Lie theorem, a generalization of this theorem. The Carathéodory - Jacobi - Lie theorem is a Theorem in Symplectic geometry which generalizes Darboux's theorem.

Notes

1. ^ Darboux (1882).
2. ^ Pfaff (1814-1815).
3. ^ Sternberg (1964) p. 140-141.
4. ^ Cf. with McDuff and Salamon (1998) p. 96.

References

• Darboux, Gaston (1882). "Sur le problème de Pfaff". Bull. Sci. Math. 6: 14-36, 49-68.  
• Pfaff, Johann Friedrich (1814-1815). "Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi". Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin: 76-136.  
• Sternberg, Shlomo (1964). Lectures on Differential Geometry. Prentice Hall.  
• McDuff, D. and Salamon, D. (1998). Introduction to Symplectic Topology. Oxford University Press. ISBN 0-19-850451-9.  

External links

• Proof of Darboux's Theorem on PlanetMath