Trigonometry of 'complex Hermitian' Type Homogeneous Symmetric Spaces

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex Hermitian' type and rank-one. The complex Hermitian elliptic CP are the generic members in this family; the remaining spaces are some contractions of the former. The method encapsulates trigonometry for this whole family of spaces into a single basic trigonometric group equation, and has 'universality' and '(self)-duality' as its distinctive traits. All previously known results on the trigonometry of CP N and CH N follow as particular cases of our general equations. The following topics are covered rather explicitly: 0) Description of the complete Cayley-Klein-Dickson family of rank-one spaces of 'complex type', 1) Derivation of the single basic group trigonometric equation, 2) Translation to the basic 'complex Hermitian' cosine, sine and dual cosine laws), 3) Comprehensive exploration of the bes-tiarium of 'complex Hermitian' trigonometric equations, 4) Uncovering of a 'Cartan' sector of Hermitian trigonometry, related with triangle symplectic area and coarea, 5) Existence conditions for a triangle in these spaces as inequalities and 6) Restriction to the two special cases of 'complex' collinear and purely real triangles. The physical Quantum Space of States of any quantum system belongs, as the complex Hermitian space member, to this parametrised family; hence its trigonometry appears as a rather particular case of the equations we obtain.