Let \(\tilde M({\mathcal U},{\tilde \Omega},\tilde g)\) be a 2m-dimensional para-Hermitian manifold [P. Libermann, Ann. Mat. Pura Appl., IV. Ser. 36, 27-120 (1954; Zbl 0056.154)] where the structure tensors (\({\mathcal U},{\tilde \Omega},\tilde g)\) are the para complex operator, the almost symplectic form and the metric tensor respectively. Let \(d\tilde p_ s\) and \(d\tilde p_{s^*}\) be the two self-orthogonal components of the line element \(d\tilde p\) of \(\tilde M.\)
The author studies the case when both \(d\tilde p_ s\) and \(d\tilde p_{s^*}\) are exterior recurrent [D. K. Patta, Tensor, NewSer. 36, 115-120 (1982; Zbl 0479.53013)]. In this case the following significant result is proved: \(\tilde M\) is endowed with a conformal symplectic structure \(CS_ p(2m;{\mathbb{R}})\), and if \(\tilde X_{\alpha}\) is the vector field of Lee associated with this structure, then \({\mathcal U}\tilde X_{\alpha}\) is an infinitesimal automorphism of \({\tilde \Omega}\). Further any such \(\tilde M({\mathcal U},{\tilde \Omega},\tilde g)\) receives a coisotropic CR-foliation [the reviewer, C. R. Acad. Sci., Paris, Sér. I 298, 149-151 (1984; Zbl 0559.53024), V. V. Goldberg and the reviewer, Int. J. Math. Math. Sci. 7, No.2, 339-350 (1984; Zbl 0553.53030)], whose leaves are vertical and mixed totally geodesic.