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 $$(\nu _pM, \Phi (p))$$ . 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.