On Complex Homogeneous Space of Vectors with Constraints

Abstract

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.