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H. Anciaux and K. Panagiotidou [1] initiated the study of non-degenerate real hypersurfaces in non-flat indefinite complex space forms in 2015. Next, in 2019 M. Kimura and M. Ortega [2] further developed their ideas, with a focus on Hopf real hypersurfaces in the indefinite complex projective space \(\mathbb CP^n_p\). In this work we are interested in the study of non-degenerate ruled real hypersurfaces in \(\mathbb CP^n_p\). We first define such hypersurfaces, then give basic characterizations. We also construct their parameterization. They are described as follows. Given a regular curve \(\alpha\) in \(\mathbb CP^n_p\), then the family of the complete, connected, complex \((n − 1)\)-dimensional totally geodesic submanifolds orthogonal to \(\alpha'\) and \(J\alpha'\), where \(J\) is the complex structure, generates a ruled real hypersurface. This representation agrees with the one given by M. Lohnherr and H. Reckziegel in the Riemannian case [3]. Further insights are given into the cases when the ruled real hypersurfaces are minimal or have constant sectional curvatures. The present results are part of a joint work together with prof. M. Ortega and prof. J.D. Pérez.

[1] H. Anciaux, K. Panagiotidou, Hopf Hypersurfaces in pseudo-Riemannian complex and para-complex space forms, Diff. Geom. Appl. 42 (2015) 1-14.
[2] M. Kimura, M. Ortega, Hopf Real Hypersurfaces in Indefinite Complex Projective, Mediterr. J. Math. (2019) 16:27.
[3] M. Lohnherr, H. Reckziegel, On ruled real hypersurfaces in complex space forms. Geom. Dedicata 74 (1999), no. 3, 267–286.

I am interested in the study of minimal isotropic Lagrangian sub manifolds \(M^n\) (\(n>2\)) in indefinite complex space forms. It is known that the dimension of \(M^n\) must satisfy \(n=3r+2\), with r a positive integer, and for \(n<14\) there exists a classification for such submanifolds. In my work I have extended the result for an arbitrary dimension n. Therefore, I have determined all the possible dimensions of \(M^n\) and found all the components of the second fundamental form, according to the metric with which \(M^n\) is endowed in each case.