Foundational result in symplectic geometry
In differential geometry, a field in mathematics, Darboux's theorem is a theorem providing a normal form for special classes of differential 1-forms, partially generalizing the Frobenius integration theorem. It is named after Jean Gaston Darboux[1] who established it as the solution of the Pfaff problem.[2]
It is a foundational result in several fields, the chief among them being symplectic geometry. Indeed, one of its many consequences is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every
-dimensional symplectic manifold can be made to look locally like the linear symplectic space
with its canonical symplectic form.
There is also an analogous consequence of the theorem applied to contact geometry.
Statement
Suppose that
is a differential 1-form on an
-dimensional manifold, such that
has constant rank
. Then
- if
everywhere, then there is a local system of coordinates
in which ![{\displaystyle \theta =x_{1}\,\mathrm {d} y_{1}+\ldots +x_{p}\,\mathrm {d} y_{p};}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- if
everywhere, then there is a local system of coordinates
in which![{\displaystyle \theta =x_{1}\,\mathrm {d} y_{1}+\ldots +x_{p}\,\mathrm {d} y_{p}+\mathrm {d} x_{p+1}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Darboux's original proof used induction on
and it can be equivalently presented in terms of distributions[3] or of differential ideals.[4]
Frobenius' theorem
Darboux's theorem for
ensures that any 1-form
such that
can be written as
in some coordinate system
.
This recovers one of the formulation of Frobenius theorem in terms of differential forms: if
is the differential ideal generated by
, then
implies the existence of a coordinate system
where
is actually generated by
.[4]
Darboux's theorem for symplectic manifolds
Suppose that
is a symplectic 2-form on an
-dimensional manifold
. In a neighborhood of each point
of
, by the Poincaré lemma, there is a 1-form
with
. Moreover,
satisfies the first set of hypotheses in Darboux's theorem, and so locally there is a coordinate chart
near
in which![{\displaystyle \theta =x_{1}\,\mathrm {d} y_{1}+\ldots +x_{m}\,\mathrm {d} y_{m}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Taking an exterior derivative now shows
![{\displaystyle \omega =\mathrm {d} \theta =\mathrm {d} x_{1}\wedge \mathrm {d} y_{1}+\ldots +\mathrm {d} x_{m}\wedge \mathrm {d} y_{m}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The chart
is said to be a Darboux chart around
.[5] The manifold
can be covered by such charts.
To state this differently, identify
with
by letting
. If
is a Darboux chart, then
can be written as the pullback of the standard symplectic form
on
:
![{\displaystyle \omega =\varphi ^{*}\omega _{0}.\,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A modern proof of this result, without employing Darboux's general statement on 1-forms, is done using Moser's trick.[5][6]
Comparison with Riemannian geometry
Darboux's theorem for symplectic manifolds 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 of coordinate differentials.
The difference is that Darboux's theorem states that
can be made to take the standard form in an entire neighborhood around
. 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.
Darboux's theorem for contact manifolds
Another particular case is recovered when
; if
everywhere, then
is a contact form. A simpler proof can be given, as in the case of symplectic structures, by using Moser's trick.[7]
The Darboux-Weinstein theorem
Alan Weinstein showed that the Darboux's theorem for sympletic manifolds can be strengthened to hold on a neighborhood of a submanifold:[8]
Let
be a smooth manifold endowed with two symplectic forms
and
, and let
be a closed submanifold. If
, then there is a neighborhood
of
in
and a diffeomorphism
such that
.
The standard Darboux theorem is recovered when
is a point and
is the standard symplectic structure on a coordinate chart.
This theorem also holds for infinite-dimensional Banach manifolds.
See also
References
- ^ Darboux, Gaston (1882). "Sur le problème de Pfaff" [On the Pfaff's problem]. Bull. Sci. Math. (in French). 6: 14–36, 49–68. JFM 05.0196.01.
- ^ Pfaff, Johann Friedrich (1814–1815). "Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi" [A general method to completely integrate partial differential equations, as well as ordinary differential equations, of order higher than one, with any number of variables]. Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin (in Latin): 76–136.
- ^ Sternberg, Shlomo (1964). Lectures on Differential Geometry. Prentice Hall. pp. 140–141. ISBN 9780828403160.
- ^ a b Bryant, Robert L.; Chern, S. S.; Gardner, Robert B.; Goldschmidt, Hubert L.; Griffiths, P. A. (1991). "Exterior Differential Systems". Mathematical Sciences Research Institute Publications. doi:10.1007/978-1-4613-9714-4. ISSN 0940-4740.
- ^ a b McDuff, Dusa; Salamon, Dietmar (2017-06-22). Introduction to Symplectic Topology. Vol. 1. Oxford University Press. doi:10.1093/oso/9780198794899.001.0001. ISBN 978-0-19-879489-9.
- ^ Cannas Silva, Ana (2008). Lectures on Symplectic Geometry. Springer. doi:10.1007/978-3-540-45330-7. ISBN 978-3-540-42195-5.
- ^ Geiges, Hansjörg (2008). An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press. pp. 67–68. doi:10.1017/cbo9780511611438. ISBN 978-0-521-86585-2.
- ^ Weinstein, Alan (1971). "Symplectic manifolds and their Lagrangian submanifolds". Advances in Mathematics. 6 (3): 329–346. doi:10.1016/0001-8708(71)90020-X.
External links
- G. Darboux, "On the Pfaff Problem", transl. by D. H. Delphenich
- G. Darboux, "On the Pfaff Problem (cont.)", transl. by D. H. Delphenich