Let $$M=G/K$$ be a Hermitian symmetric space of noncompact type. We provide way constructing K-equivariant embeddings from M to its tangent $$T_oM$$ at the origin by using polarity K-action. As an application, we reconstruct holomorphic embedding so called Harish-Chandra realization and symplectomorphism constructed Di Scala-Loi Roos under appropriate identifications spaces. Moreover, character...

Pre-symmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A structural projection on a pre-symmetric space A∗ represents an operation on the corresponding system, and has as its range a further pre-symmetric space which represents the state space of the resulting system and symmetries of the system are represented by elements of the group Aut(A∗) of ...

The notion of a Γ-symmetric space is a generalization of the classical notion of a symmetric space, where a general finite abelian group Γ replaces the group Z 2. The case Γ = Z k has also been studied, from the algebraic point of view by V.Kac [10] and from the point of view of the differential geometry by Ledger, Obata [12], Kowalski [11] or Wolf-Gray [18] in terms of k-symmetric spaces. In t...

The notion of a Γ-symmetric space is a generalization of the classical notion of a symmetric space, where a general finite abelian group Γ replaces the group Z2. The case Γ = Z k has also been studied, from the algebraic point of view by V.Kac [14] and from the point of view of the differential geometry by Ledger, Obata [17], Kowalski [16] or Wolf-Gray [20] in terms of k-symmetric spaces. In th...

These notes cover the basics of Riemannian geometry, Lie groups, and symmetric spaces. This is just a listing of the basic definitions and theorems with no in-depth discussion or proofs. Some exercises are included at the end of each section to give you something to think about. See the references cited within for more complete coverage of these topics. Many geometric entities are representable...