A homogeneous space $\mathcal{V}$ of complex constrained vectors in $\mathbb{C}^3,$ representing complex velocities is introduced. The corresponding representation of the complex special orthogonal group of transformations acting on $\mathcal{V}$ is also examined. The requirement for real vector magnitudes is addressed by imposing orthogonality between the real and the imaginary parts of vectors and use of the non-conjugate scalar product. We present the orthogonal transformations acting on $\mathcal{V}$ in terms of the polar decomposition of complex orthogonal matrices. The group link problem and the homogeneity of the space $\mathcal{V}$ are also discussed. Finally, we briefly consider the convenience of the space $\mathcal{V}$ in theoretical calculations.

The notion of ideal embeddings was introduced by the author at the Third Pacific Rim Geometry Conference held at Seoul in 1996. Roughly speaking, an ideal embedding is an isometrical embedding which receives the least possible amount of tension from the surrounding space at each point. In this article, we study ideal embeddings of irreducible compact homogenous spaces in Euclidean spaces. Our main result states that if $\pi: M\to N$ is a covering map between two irreducible compact homogeneous spaces with $\lambda_1(M)\ne \lambda_1(N)$, then $N$ does not admit an ideal embedding in a Euclidean space, although $M$ could.

Adler had showed that the Toda system can be given a coadjoint orbit description. We quantize the Toda system by viewing it as a single orbit of a multiplicative group of lower triangular matrices of determinant one with positive diagonal entries. We get a unitary representation of the group with square integrable polarized sections of the quantization as the module. We find the Rawnsley coherent states after completion of the above space of sections. We also find non-unitary finite dimensional quantum Hilbert spaces for the system. Finally we give an expression for the quantum Hamiltonian for the system.

We calculate the Chern numbers of ${\rm SU}(2)$-homogeneous Einstein-Maxwell gravitational instantons with boundary at infinity. By restating these numbers as Chern-Simons invariants on the boundary apparent conflicting results emerge. We resolve this issue examining the topological stability of the self-gravitating Abelian fields. No quantization carrying physical meaning is found when the background is a Taub-NUT space. However the magnetic charge of dyons on Taub-Bolt spaces is found to be of topological quantum nature. In this framework electric charge is quantized by a consistency condition.

It is a crucial problem in robotics field to cage an object using robots like multifingered hand. However the problem what is the caging for general geometrical objects and robots has not been well-described in mathematics though there were many rigorous studies on the methods how to cage an object by certain robots. In this article, we investigate the caging problem more mathematically and describe the problem in terms of recursion of the simple euclidean moves. Using this description, we show that the caging has the degree of difficulty which is closely related to a combinatorial problem and a wire puzzle. It implies that in order to capture an object by caging, from a practical viewpoint the difficulty plays an important role.

In present paper we define and study $n$-characteristic vector fields. We present definition of Tanaka-Webster connection, then use it for studying the behavior of $n$-characteristic vector fields. Also we show some results about of these vector fields by Tanaka-Webster connection.

We describe a curious relationship between orthogonal spheres and causal relationships in Minkowski space.

In the present paper we demonstrate that the massless fields can be described by two types of potentials with different space-time properties and different Lorentz transformations. In particular, we discuss the possible applications of such approach to the description of electromagnetic field and weak gravity.