Generalized quasi-statistical structures (Q2233215)

A quasi-statistical manifold is a manifold \(M\) endowed with a pseudo-Riemannian metric \(h\) and a connection \(\nabla\) such that \[ (\nabla_X h)(Y,Z)-(\nabla_Y h)(X,Z)+h(T^{\nabla}(X,Y) , Z)=0, \] for all vector fields \(X,Y,Z\), where \(T^{\nabla}\) is the torsion of the connection \(\nabla\). In this case, one says that \((h,\nabla )\) is a quasi-statistical structure on \(M\). In this paper the authors, given a pseudo-Riemannian metric \(h\) and a connection \(\nabla\) on a manifold \(M\), introduce a generalized complex structure and generalized product structure on the generalized tangent bundle \(TM\oplus T^*M\). Then, they show that they are \(\nabla\)-integrable if and only if \((h,\nabla )\) is a quasi-statistical structure. Further, the authors introduce the notion of generalized quasi-statistical structure. They show that a quasi-statistical structure on \(M\) yields two generalized quasi-statistical structures on \(TM\oplus T^*M\). They introduce a dual quasi-statistical connection and study its properties. The results are described in terms of Patterson-Walker and Sasaki metrics on \(T^*M\) and \(TM\). Moreover, the prolongations of quasi-statistical structures on \(M\) to its cotangent and tangent bundles are studied. Finally, the authors define Norden and para-Norden structures on \(T^*M\) and \(TM\).