Legendre manifold

An $n$-dimensional smooth submanifold $L^n$ of a $(2n + 1)$- dimensional contact manifold $M^{2n+1}$ (that is, a manifold endowed with a Pfaffian form $\alpha $ such that the exterior product of it with the $n$- th exterior power of its exterior differential $ \alpha \wedge ( d \alpha ) ^ {n} \neq 0 $ at all points of $ M ^ {2n+1} $), such that the Pfaffian form $ \alpha $ that specifies the contact structure on $ M ^ {2n+1} $ vanishes identically on $ L ^ {n} $( that is, $ \alpha ( X) = 0 $ for any vector $ X $ that is tangent to $ L ^ {n} $ at some point of $ L ^ {n} $). In the important special case when $ M ^ {2n+1} = \mathbf R ^ {2n+1} $ with coordinates $ ( p _ {1} \dots p _ {n} , q _ {1} \dots q _ {n} , r ) $, $ \alpha = \sum_{i=1}^ {n} p _ {i} dq _ {i} - dr $ and $ L ^ {n} $ is situated so that the $ q _ {i} $ can be taken as coordinates on it, the condition that $ L ^ {n} $ is a Legendre manifold means that it is specified by equations of the form

$$ r = f ( q _ {1} \dots q _ {n} ) ,\ p _ {1} = \frac{\partial f }{ \partial q _ {1} } \dots p _ {n} = \frac{\partial f }{\partial q _ {n} } . $$

If the $ p _ {i} $ can also be taken as coordinates on $ L ^ {n} $, then the coordinates $ q _ {i} $ and $ p _ {i} $ are connected by a Legendre transformation (cf. Legendre transform); if this cannot be done in a neighbourhood of some point, then the Legendre transformation has a singularity at this point.

Examples of Legendre manifolds occurred long ago in various questions of analysis and geometry, but the idea of the Legendre manifold itself was introduced comparatively recently by analogy with a Lagrangian manifold.

Comments

The generalization of solutions of first-order partial differential equations to Legendre manifolds is due to S. Lie, see [a1], §23, 26, although Lie did not give a name to it.

References

[1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) Zbl 0692.70003 Zbl 0572.70001 Zbl 0647.70001
[2]	V.I. [V.I. Arnol'd] Arnold, A.B. [A.B. Givent'al] Giventhal, "Symplectic geometry" , Dynamical Systems , IV , Springer (1989) pp. Chapt. 4 (Translated from Russian) MR1866631 MR1768639 MR1356438 MR0842908 MR0820079 Zbl 1048.00010 Zbl 1021.53053 Zbl 0973.53501
[a1]	V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 1 , Birkhäuser (1985) pp. Chapt. 20 (Translated from Russian) MR777682 Zbl 0554.58001
[a2]	S. Lie, F. Engel, "Theorie der Transformationsgruppen" , II , Leipzig (1930) MR1510035 Zbl 50.0002.01 Zbl 53.0026.07 Zbl 25.0626.01 Zbl 25.0623.01 Zbl 23.0376.01 Zbl 22.0372.01 Zbl 23.0364.01 Zbl 20.0368.01 Zbl 21.0356.02 Zbl 16.0326.01 Zbl 12.0292.01 Zbl 11.0258.02 Zbl 10.0260.01 Zbl 10.0258.01