On Symmetric Cauchy–Riemann Manifolds

Abstract

The Riemannian symmetric spaces play an important role in different branches of mathematics. By definition, a (connected) Riemannian manifold M is called symmetric if, to every a ∈ M , there exists an involutory isometric diffeomorphism sa:M → M having a as isolated fixed point in M (or equivalently, if the differential dasa is the negative identity on the the tangent space Ta = TaM of M at a). In case such a transformation sa exists for a ∈ M , it is uniquely determined and is the geodesic reflection of M about the point a. As a consequence, for every Riemannian symmetric space M , the group G = GM generated by all symmetries sa, a ∈ M , is a Lie group acting transitively on M . In particular, M can be identified with the homogeneous space G/K for some compact subgroup K ⊂ G. Using the elaborate theory of Lie groups and Lie algebras E.Cartan classified all Riemannian symmetric spaces.