Extrinsic symplectic symmetric spaces

Abstract

We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also build a natural star-quantization on a class of examples

Extrinsically immersed symplectic symmetric spaces

“…Remark The reader who is familiar with [1] will note that dı = + τ is an extrinsic symplectic morphism of g if and only if satisfies the conditions in [1, Lemma 1.5] when identifying p ∼ = W 1 by the isomorphism τ .…”

“…In this language [1, Lemmas 1.5, 1.7] as well as the equivalence of equations (1.14) and (1.24) in [1] yield the following. Proposition 3.6 Let g = k ⊕ p be a symplectic symmetric pair with the ad k -invariant symplectic form ω ∈ 2 p * , let R = i, j a i j A i ∧ A j ∈ 2 sp(p, ω) be its curvature and let κ i j = κ ji ∈ k * be as in (5).…”

“…For instance, a symplectic vector space, regarded as a flat symmetric space, can be embedded non-affinely into a higher-dimensional vector space as an e.s.s.s. [1]. It is precisely this non-uniqueness which we want to illuminate.…”

“…In [1], extrinsic symplectic symmetric spaces (e.s.s.s.) were considered for the first time, and their basic algebraic and geometric properties were described.…”

On the solvability of the transvection group of extrinsic symplectic symmetric spaces

“…In [6], extrinsic symplectic symmetric spaces (e.s.s.s.) were considered for the first time, and their basic algebraic and geometric properties were described.…”