Kaup, Wilhelm; Zaitsev, Dmitri 'On symmetric Cauchy-Riemann manifolds' in Advances in Mathematics, 149, (2), 2000, pp 145 - 181
Advances in Mathematics 149 2
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.
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