Lagrangian submanifolds in Hyperkahler manifolds, Legendre transformation

拉格朗日子流形与辛群胚附近的超凯勒度量

Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids

本文将 Feix-Kaledin 定理推广到复拉格朗日子流形邻域，把超凯勒结构的存在性归结为全纯辛结构的形变问题，并证明凯勒型全纯泊松曲面上的全纯辛群胚在单位截面附近具有超凯勒结构。

Infinitesimal deformations and stabilities of singular Legendre submanifolds

We give the characterization of Arnol'd-Mather type for stable singular Legendre immersions. The most important building block of the theory is providing a module structure on the space of infinitesimal integral deformations by means of the notion of natural liftings of differential systems and of contact Hamiltonian vector fields.

Riemannian geometry over different normed division algebra

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkahler and G_2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Simila…

Instantons and branes in manifolds with vector cross product

In this paper we study the geometry of manifolds with vector cross product and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and coassociative submanifolds in G_2-manifolds. Second we classify Kahler manifolds with the complex analog of vector cross product, namely they are Calabi-Yau manifolds and hyperkahler manifolds. Furthermore we study instantons, Neumann branes and Dirichlet branes on these manifolds.…

The Maslov cycle as a Legendre singularity and projection of a wavefront set

A Maslov cycle is a singular variety in the lagrangian grassmannian L(V) of a symplectic vector space V consisting of all lagrangian subspaces having nonzero intersection with a fixed one. Givental has shown that a Maslov cycle is a Legendre singularity, i.e. the projection of a smooth conic lagrangian submanifold S in the cotangent bundle of L(V). We show here that S is the wavefront set of a Fourier integral distribution which is "evaluation at 0 of the quantizations".

Contact Dynamics versus Legendrian and Lagrangian Submanifolds

We are proposing Tulczyjew's triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two d…