A characterization of oriented Grassmann manifolds
The geodesic spheres in two-point homogeneous spaces are hypersurfaces whose second fundamental forms have some nice properties. T. J. Willmore and the reviewer have used these properties to characterize, at least locally, the two-point homogeneous spaces. Jacobi vector fields are used extensively in their work. It is natural to ask if the other symmetric spaces can be characterized locally in a similar way. This paper is a first step in answering this problem. The authors consider a special class of symmetric spaces, namely the real Grassmann manifolds and their compact duals. A characterization is given in terms of a special parallel tensor field of type (1,3) satisfying several additional algebraic properties and with a special relation with the second fundamental form of the geodesic spheres. These results include some of the results of T. J. Willmore and the reviewer as a special case. This paper has been extended by the second author [Indian J. Pure Appl. Math. 15, 99-112 (1984; Zbl 0531.53036)] to the complex Grassmann manifolds and a unified theory for all the Grassmannians will be published in one of his forthcoming papers.
scientific article; zbMATH DE number 3872204