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.
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 |