Riemannian and spectral geometry

 Riemannian and spectral geometry. Recall that the differentiable structure of a smooth manifold M gives rise to a canonical symplectic form on its cotangent

g

⊂

g

bundle T ∗M . Giving a Riemannian metric g on M is equivalent to prescribing its unit cosphere bundle S ∗M T ∗M , and the restriction of the canonical 1-form from T ∗M gives S ∗M the structure of a contact manifold. The Reeb flow on S ∗M is the geodesic flow (free particle motion).

· ·

In a somewhat different direction, each symplectic form ω on some manifold M distinguishes the class of Riemannian metrics which are of the form ω(J , ) for some almost complex structure J .

These (and other) connections between symplectic and Riemannian geometry are by no means completely explored, and we believe there is still plenty to be discovered here. Here are some examples of known results relating Riemannian and symplectic aspects of geometry.

Lagrangian submanifolds. A middle-dimensional submanifold L of (M, ω)

is called Lagrangian if ω vanishes on T L.

1. Volume. Endow complex projective space ¢Pn with the usual Ka¨hler metric and the usual Ka¨hler form. The volume of submanifolds is taken with respect to this Riemannian metric. According to a result of Givental–Kleiner–Oh, the standard RPn in ¢Pn has minimal volume among all its Hamiltonian deformations [74]. A partial result for the Clifford torus in ¢Pn can be found in [38]. The torus S1 × S1 ⊂ S2 × S2 formed by the equators is also volume minimizing among its Hamiltonian deformations, [50]. If L is a closed Lagrangian

   submanifold of R2n, ω0 , there exists according to [98] a constant C depending on L such that

   
   

   vol (φH (L)) ≥ C for all Hamiltonian deformations of L. (1–1)

2. Mean curvature. The mean curvature form of a Lagrangian submanifold L in a Ka¨hler–Einstein manifold can be expressed through symplectic invariants of L, see [15].

The first eigenvalue of the Laplacian. Symplectic methods can be used to estimate the first eigenvalue of the Laplace operator on functions for certain Riemannian manifolds [80].

Short billiard trajectories. Consider a bounded domain U ⊂ Rn with smooth boundary. There exists a periodic billiard trajectory on U of length l with

ln ≤ Cn vol(U ) (1–2)

where Cn is an explicit constant depending only on n, see [98; 30].