For this quarter of century, quantum differential operators in a lower dimensional submanifold embedded or immersed in real n-dimensional euclidean space E n have been studied as physical models, which are realized as restriction of the operators in E n to the submanifold. For this decade, I have been investigating the Dirac operators in the submanifold, which are identified with operators of t...

The complexity in a biological system may be caused by both the number of variables involved and the number of system constants that can vary. A biological system in the subcellular level often stabilizes after a certain period of time. Its asymptote can then be described as an equilibrium under certain continuity assumptions. The biological quantities at the equilibrium can be detected by expe...

We introduce an invariant, called the contact number, associated with each Euclidean submanifold. We show that this invariant is, surprisingly, closely related to the notions of isotropic submanifolds and holomorphic curves. We are able to establish a simple criterion for a submanifold to have any given contact number. Moreover, we completely classify codimension-2 submanifolds with contact num...

We revisit the Lagrange and Delaunay systems of equations for the orbital elements, and point out a previously neglected aspect of these equations: in both cases the orbit resides on a certain 9-dimensional submanifold of the 12-dimensional space spanned by the orbital elements and their time derivatives. We demonstrate that there exists a vast freedom in choosing this submanifold. This freedom...

A submanifold in a symmetric space is called equifocal if it has a globally flat abelian normal bundle and its focal data is invariant under normal parallel transportation. This is a generalization of the notion of isoparametric submanifolds in Euclidean spaces. To each equifocal submanifold, we can associate a Coxeter group, which is determined by the focal data at one point. In this paper we ...

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...

Advanced metering infrastructure (AMI) is a key part of the smart grid; thus, one of the most important concerns is to offer a secure mutual authentication.  This study focuses on communication between a smart meter and a server on the utility side. Hence, a mutual authentication mechanism in AMI is presented based on the elliptic curve cryptography (ECC) and one time signature (OTS) consists o...