This article is an excellent part of a survey work carried out by the authors. The approach is from general to particular, starting from the geometry of almost complex manifolds and then specializing to the case of homogeneous spaces of Lie groups so as to realize Cartan classical domains as particular ones. These are, in fact, irreducible Hermitian symmetric spaces of the noncompact type. Cartan classical domains are classified into four series which are quotients of non-compact groups by compact groups, i.e. \({\mathcal A}(p,q)=SU(p,q)/S(U(p)\times U(q)),{\mathcal B}(n)=SO^*(2n)/U(n)\), \(\mathcal C(n)=Sp(2n,\mathbb{R})/U(n)\), \({\mathcal D}(n)=SO(n,2)/SO(n)\times SO(2)\), and two exceptional ones: \(E_ 6/SO(10)\times U(1)\) and \(E_ 7/E_ 6\times U(1)\). The spaces \({\mathcal A}(n)={\mathcal A}(n,n)\) are \(2n^ 2\)-dimensional and are called “non- compact complex Grassmannian”, the \(\mathcal C(n)\) are called “Siegel half planes”, while the \({\mathcal D}(n)\) are \(2n\)-dimensional and are called “Lie balls”. Low dimensional isomorphisms of Lie groups belonging to different series imply isomorphisms between the lowest members of the above families, e.g. the 2-dimensional domain \({\mathcal D}(1)=SO(1,2)/SO(2)=SL(2,\mathbb{R})/SO(2)={\mathcal A}(1)=U(1,1)/U(1)=\mathcal C(1)=Sp(2,\mathbb{R})/U(1)\) is the familiar unit disc. The 4-dimensional domain \({\mathcal D}(2)=SO(2,2)/SO(2)\times SO(2)\) is the only one which is not irreducible. The authors mainly deal with the \({\mathcal A}(n)\) and \({\mathcal D}(n)\) series and take either \({\mathcal A}(1)={\mathcal D}(1)\) (the unit disc) or \({\mathcal A}(2)={\mathcal D}(4)\) (the 8-dimensional Lie-ball) for all explicit examples in their studies.
The authors first discuss the boundaries of Cartan classical domains, global and local charts theorem, the Cayley transformation for the \(\mathcal A\) and \(\mathcal D\) series, matrix realizations of Cartan domains, geometrical aspects of Bergman-Szegö-kernels and then study wavelets and relativistic wavelets analysis on Cartan domains, Lie balls and actions of the conformal group, conformal transformations of spacetime, Riemannian geometry of \({\mathcal D}(n)\), relativistic phase spaces, the Poincaré-Cartan momentum map, conformal field theory, Weyl-Berezin calculus on Cartan domains, quantum mechanics in arbitrary frames, harmonicity cells, Lie spheres and Lie balls, the Lelong map and its physical meaning etc.
All the important results on the geometry of complex manifolds have been delineated so beautifully and well knitted that it exhibits a high craftmanship of writing mathematical articles, especially a pretty long list of research papers in the references is very useful for both mathematicians as well as physicists.