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Paracomplex structures and affine symmetric spaces. (English) Zbl 0585.53029
An almost paracomplex structure on a 2n-dimensional smooth manifold M is a smooth (1,1) tensor field I satisfying i) $$I^ 2=id$$, ii) for each $$p\in M$$, the $$\pm 1$$ eigenspaces $$T_ p^{\pm}(M)$$ of $$I_ p$$ are both n-dimensional subspaces of $$T_ p(M)$$. If the tensor $$T(X,Y)=[IX,IY] - I[IX,Y] - I[X,IY] + [X,Y]$$ vanishes identically on M then (M,I) is called a paracomplex manifold.
The main purpose is to develop the theory of paracomplex manifolds in parallel with the theory of complex manifolds. The authors introduce a paracomplex analogue of Hermitian symmetric spaces, called parahermitian symmetric spaces, giving the infinitesimal classification when the automorphism group is semisimple.
Reviewer: I.Dotti Miatello

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C35 Differential geometry of symmetric spaces
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