A ruled real hypersurface in a complex space form is a real hypersurface having a codimension one foliation by totally geodesic complex hyperplanes of the ambient space. Our main purpose of this paper is to introduce a new viewpoint to investigate such hypersurfaces in complex hyperbolic space 
$$\mathbb {CH}^n$$

. As an application, we study minimal ruled real hypersurfaces in 
$$\mathbb {CH}^n$$

 and also those having constant scalar curvature.

New construction of ruled real hypersurfaces in a complex hyperbolic space and its applications

We describe all centrioles in irreducible simply connected pointed symmetric spaces of compact type in terms of the root system of the ambient space, and we study some geometric properties of centrioles.

/pdf/centrioles-in-symmetric-spaces-ky3wngam4q.pdf

Centrioles in symmetric spaces

We introduce a new technique to the study and identification of submanifolds of simply-connected symmetric spaces of compact type based upon an approach computing $k$-positive Ricci curvature of the ambient manifolds and using this information in order to determine how highly connected the embeddings are. This provides codimension ranges in which the Cartan type of submanifolds satisfying certain conditions which generalize being totally geodesic necessarily equals the one of the ambient manifold. Using results by Guijarro--Wilhelm our approach partly generalizes recent work by Berndt--Olmos on the index conjecture.

The flavour of intermediate Ricci and homotopy when studying submanifolds of symmetric spaces

Existence and uniqueness for a class of cauchy-problems with characteristic initial manifolds

In Riemannian geometry the study of minimal submanifolds has given the most important, higher-dimensional generalizations of geodesies. Especially significant from a global point of view are closed minimal submanifolds (generalizing closed geodesies); these raise many hard problems. In this paper we study existence and uniqueness questions in the case of the simplest topological type; i.e. minimal hyperspheres. We restrict ourselves to study such questions for the compact two-point homogeneous spaces; these spaces constitute the most natural generalization of classical (three-point homogeneous) spherical geometry. They can be characterized equivalently as (i) compact two-point homogeneous spaces, (ii) compact rank 1 symmetric spaces, or (iii) irreducible compact positively curved symmetric spaces. Since the standard spheres have been investigated in great detail in connection with the "Spherical Bernstein Problem", we only consider the complex projective spaces CP(n), the quaternionic projective spaces HP(n), and the Cayley projective plane Ca(2) here.

https://msp.org/pjm/1996/173-1/pjm-v173-n1-p15-s.pdf

Minimal hyperspheres in two-point homogeneous spaces

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1988, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/on-the-existence-of-minimal-hyperspheres-in-compact-eoqv266v5k.pdf

On the existence of minimal hyperspheres in compact symmetric spaces

/pdf/on-minimal-immersions-of-s-n-1-into-s-n-1-n-ge-4-3p7wp2ao1r.pdf

On minimal immersions of $S^{n-1}$ into $S^n(1),\;n \ge 4$

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On the existence of minimal hyperspheres in compact symmetric spaces

On minimal immersions of $S^{n-1}$ into $S^n(1),\;n \ge 4$