A Lagrangian submanifold of a symplectic manifold 
is an 
-dimensional submanifold 
on which the restriction of 
vanishes. For example, the zero
section of the cotangent bundle 
is Lagrangian for its canonical symplectic
form 
.
Here the canonical 1-form is 
, where 
is the bundle projection. A Lagrangian submanifold
in this cotangent bundle is called exact if 
for a smooth real
function 
on 
.
The graph of 
is an exact Lagrangian submanifold. The nearby
Lagrangian conjecture concerns exact embedded examples that are compact
without boundary.
Lagrangian Submanifold
See also
Cotangent Bundle, Lagrangian Fibration, Nearby Lagrangian Conjecture, Symplectic Form, Symplectic ManifoldExplore with Wolfram|Alpha
References
Cannas da Silva, A. Lectures on Symplectic Geometry. Berlin: Springer-Verlag, 2001. https://people.math.ethz.ch/~acannas/Papers/lsg.pdf.Cite this as:
Weisstein, Eric W. "Lagrangian Submanifold." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LagrangianSubmanifold.html