NORMAL HOLONOMY AND RATIONAL PROPERTIES OF THE SHAPE OPERATOR

CARLOS ENRIQUE OLMOS Y RICHAR RIAÑO-RIAÑO

MANUSCRIPTA MATHEMATICA

SPRINGER

Año: 2018 vol. 157 p. 467 - 482

0025-2611

Let $M$ be a most singular orbit of the isotropy representation of a simple symmetric space. Let $(u _i, Phi _i)$ be an irreducible factor of the normal holonomy representation $(u _pM, Phi (p))$. We prove that there exists a basis of a section $Sigma _isubset u _i$ of $Phi _i$ such that the corresponding shape operators have rational eigenvalues (this is not in general true for other isotropy orbits). Conversely, this property, if referred to some non-transitive irreducible normal holonomy factor, characterizes the isotropy orbits. We also prove that the definition of a submanifold with constant principal curvatures can be given by using only the traceless shape operator, instead of the shape operator, restricted to a non-transitive (non necessarily irreducible) normal holonomy factor. This article generalizes previous results of the authors that characterized Veronese submanifolds in terms of normal holonomy.