The starting point for this paper is the paper of the third author [Ann. Mat. Pura Appl. (4) 194, No. 5, 1505–1525 (2015; Zbl 1325.53106)], where the concept of interpolation of supplementary compatible geometric structures is introduced. In the case of a complex manifold \((M,j)\) the third author obtained six families of distinguished generalized complex or paracomplex structures on \(M\). Each of them interpolates between two geometric structures on \(M\) compatible with \(j\), for instance, between totally real foliations and Kähler structures, or between hypercomplex and \(\mathbb{C}\)-symplectic structures. A similar analysis was carried out for a symplectic manifold \((M,\omega)\) obtaining the notion of a structure generalizing \(\mathbb{C}\)-symplectic structures and bi-Lagrangian foliations.
In the present paper the authors obtain similar results, starting from a pseudo-Riemannian manifold \((M,g)\). First, they recall the definitions and properties of generalized complex or paracomplex structures. They present some geometric structures on \(M\) compatible with \(g\) such as \((\lambda,0)\)- or \((0,l)\)-structures, with \(\lambda, l=\pm 1\). For example, in the cases \(\lambda =-1\) and \(l=-1\), one obtains the anti-Hermitian and almost pseudo-Kähler structures, respectively. Next, one can define families of generalized complex or paracomplex structures on \(M\), called integrable \((\lambda, l)\)- structures which, in a certain sense, interpolate between integrable \((\lambda,0)\)- and \((0,l)\)-structures on \(M\). For a bilinear form \(c\) on a real vector space \(V\), one denotes by \(c^\flat\in \mathrm{End}(V,V^*)\) the endomorphism defined by \(c^\flat(u)(v)=c(u,v)\). The form \(c\) is symmetric (respectively, skew-symmetric) if \((c^\flat)^*=c^\flat\) (respectively \((c^\flat)^*=-c^\flat\)). Considering a pseudo-Riemannian manifold \((M,g)\), for \(k=-1\) or \(k=1\), one denotes by \( I_k\) the complex, respectively paracomplex structure on \(TM\) given by \( I_k=\left(\begin{matrix} 0 & k(g^\flat)^{-1}\cr g^\flat & 0\end{matrix}\right)\). A generalized complex structure \(S\) (for \(\lambda =-1\)) or a generalized paracomplex structure \(S\) (for \(\lambda =1)\) is an integrable \((\lambda, l)\)-structure on \((M,g)\) if \(SI_k=-I_kS\), where \(k=-\lambda l\).
The authors prove Theorem 3.6: Consider a pseudo-Riemannian manifold \((M,g)\). Then, for \(\lambda =\pm 1,\;l=\pm 1\), the integrable \((\lambda, l)\)-structures on \((M,g)\) interpolate between integrable \((\lambda,0)\)- and \((0,l)\)-structures on \((M,g)\). Next they obtain the explicit form of the integrable \((\lambda,l)\)-structure \(S\) on the Riemannian manifold \((M,g)\). Moreover, the authors obtain the signature associated to an integrable \((1,1)\)-structure on \((M,g)\). Next they introduce the integrable \((1,1;n)\)-structures on \((M,g)\). In the next section, the authors consider the \((\lambda,l)\)-twistor bundles over \((M,g)\). They interpret the integrable \((\lambda,l)\)-structures on the pseudo-Riemannian manifold \((M,g)\) of dimension \(m\) and signature \((p,q)\), as smooth sections in a certain fiber bundle with fiber \(G/H\), where the group \(G\) and its subgroup \(H\) are defined according to the type of the considered \((\lambda,l)\)-structure. Finally, the authors study integrable \((-1,-1)\)-structures on nilmanifolds.