Legendre manifold

An -dimensional smooth submanifold of a -dimensional contact manifold (that is, a manifold endowed with a Pfaffian form such that the exterior product of it with the -th exterior power of its exterior differential at all points of ), such that the Pfaffian form that specifies the contact structure on vanishes identically on (that is, for any vector that is tangent to at some point of ). In the important special case when with coordinates , and is situated so that the can be taken as coordinates on it, the condition that is a Legendre manifold means that it is specified by equations of the form

If the can also be taken as coordinates on , then the coordinates and 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.

References

[1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[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)

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

[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)
[a2]	S. Lie, F. Engel, "Theorie der Transformationsgruppen" , II , Leipzig (1930)