In this paper we define an invariant of a pair of 6 dimensional symplectic manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of the bounding cochains (solution of A infinity version of Maurer-Cartan equation of the filtered A infinity algebra associated to the Lagrangian su...

We show every toric Sasaki–Einstein manifold S admits a special Legendrian submanifold L which arises as the link fix(τ) ∩ S of the fixed point set fix(τ) of an anti-holomorphic involution τ on the cone C(S). In particular, we obtain a special Legendrian torus S × S in an irregular toric Sasaki–Einstein manifold which is diffeomorphic to S × S. Moreover, there exists a special Legendrian subman...

This is the content of a talk given by the author at the 2009 Lehigh University Geometry/Topology Conference. Using the definition of connection given by Dieudonné, the Sasaki metric on the tangent bundle to a Riemannian manifold is expressed in a natural way. Also, the following property is established. The induced metric on the tangent bundle of an isometrically embedded submanifold is the Sa...

Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox’s theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the ∂connected sum decomposition of th...

We introduce the notion of the projective linking number LinkP(Γ, Z) of a compact oriented real submanifold Γ of dimension 2p − 1 in complex projective n-space P with an algebraic subvariety Z ⊂ P − Γ of codimension p. This notion is related to projective winding numbers and quasi-plurisubharmonic functions, and it is generalized to any projective manifold X . It is shown that a basic conjectur...

Moduli spaces M of self-dual SU(2) connections (“instantons”) over a compact Riemannian 4.manifold (hl, g) carry a natural L2 metric g, which is generally incomplete. For instantons of Pontryagin index 1 over a compact, simply connected, oriented, positive-definite base manifold, the completion M is Donaldson’s compactification; in fact the boundary of the completion is an isometric copy of (M,...

We deal with a CR-submanifold M of a para-Kählerian manifold M, which carries a J-skew-symmetric vector field X. It is shown that X defines a global Hamiltonian of the symplectic form Ω on M and JX is a relative infinitesimal automorphism of Ω. Other geometric properties are given. 1. Introduction. CR-submanifolds M of some pseudo-Riemannian manifolds M have been first investigated by Rosca [10...

Wilson loops in N = 4 supersymmetric Yang-Mills theory correspond at strong coupling to extremal surfaces in AdS5. We study a class of extremal surfaces known as special Legendrian submanifolds. The ”hemisphere” corresponding to the circular Wilson loop is an example of a special Legendrian submanifold, and we give another example. We formulate the necessary conditions for the contour on the bo...

We study the task to infer and to track the viewpoint onto a 3D rigid object by observing its image contours in a sequence of images. To this end, we consider the manifold of invariant planar contours and learn the low-dimensional submanifold corresponding to the object contours by observing the object off-line from a number of different viewpoints. This submanifold of object contours can be pa...