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.