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In this work, we are interested in a non symmetric homogeneous space, namely$SO(2m)/Sp(m)$. We show that this space admits a structure of $Z_2^2$-symmetricspace. We describe all the non degenerated metrics and classify the Riemannianand Lorentzian ones.

Riemannian and Lorentzian structures on the non symmetric space SO(2m)/Sp(m)