A dual pair for free boundary fluids

We construct a dual pair associated to the Hamiltonian geometric formulation of perfect fluids with free boundaries. This dual pair is defined on the cotangent bundle of the space of volume preserving embeddings of a manifold with boundary into a boundaryless manifold of the same dimension. The dual pair properties are rigorously verified in the infinite-dimensional Fréchet manifold setting. It provides an example of a dual pair associated to actions that are not completely mutually orthogonal.

ALMOST COMPLEX STRUCTURE ON PATH SPACE

Let M be a complex manifold and let PM ≔ C∞([0, 1], M) be space of smooth paths over M. We prove that the induced almost complex structure on PM is weak integrable by extending the result of Indranil Biswas and Saikat Chatterjee of [Geometric structures on path spaces, Int. J. Geom. Meth. Mod. Phys.8(7) (2011) 1553–1569]. Further we prove that if M is smooth manifold with corner and N is any complex manifold then induced almost complex structure 𝔍 on Fréchet manifold C∞(M, N) is weak integrable.