Normal holonomy and rational properties of the shape operator

Abstract

Let $M$ be a most singular orbit of the isotropy representation of a simple symmetric space. Let $({\nu }_i,{\Phi }_i)$ be an irreducible factor of the normal holonomy representation $({\nu }_{p}M,\Phi (p))$. We prove that there exists a basis of a section ${\Sigma }_i\subset {\nu }_i$ 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.

15 pages