The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.
be a closed (compact without boundary) symplectic manifold.
, the symplectic form
induces a Hamiltonian vector field
defined by the formula The function
is called a Hamiltonian function.
Suppose there is a smooth 1-parameter family of Hamiltonian functions
This family induces a 1-parameter family of Hamiltonian vector fields
The family of vector fields integrates to a 1-parameter family of diffeomorphisms
is a called a Hamiltonian diffeomorphism of
The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of
is greater than or equal to the number of critical points of a smooth function on
be a closed symplectic manifold.
is called nondegenerate if its graph intersects the diagonal of
For nondegenerate Hamiltonian diffeomorphisms, one variant of the Arnold conjecture says that the number of fixed points is at least equal to the minimal number of critical points of a Morse function on
, called the Morse number of
In view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a field
The weak Arnold conjecture says that for
a nondegenerate Hamiltonian diffeomorphism.
[2][3] The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, gives a lower bound on the number of intersection points of two Lagrangian submanifolds L and
in terms of the Betti numbers of
intersects L transversally and
-dimensional symplectic manifold, let
be a compact Lagrangian submanifold of
be an anti-symplectic involution, that is, a diffeomorphism
, whose fixed point set is
be a smooth family of Hamiltonian functions on
by flowing along the Hamiltonian vector field associated to
intersects transversely with
, then The Arnold–Givental conjecture has been proved for several special cases.