Abstract
We consider a homogeneous manifold \(\mathcal{H}\) embedded in \(\mathbb{C}^3\) composed of complex vectors with constraints, potentially representing space of complex velocities. The imposed constraints include orthogonality between the real and the imaginary parts of vectors which together with the non-conjugate scalar product provide real vector magnitudes. The corresponding representation of the group SO\((3,\mathbb{C})\) acting on \(\mathcal{H}\) which is in agreement to the polar decomposition of complex orthogonal matrices is also discussed.
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