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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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