# Normal holonomy and rational properties of the shape operator

## TL;DR

This paper investigates the rational eigenvalues of shape operators in singular orbits of symmetric spaces, providing new characterizations of isotropy orbits and submanifolds with constant principal curvatures based on holonomy properties.

## Contribution

It establishes the existence of rational eigenvalues for shape operators in certain normal holonomy factors and characterizes isotropy orbits via this property, generalizing previous results.

## Key findings

- Existence of rational eigenvalues for shape operators in specific holonomy factors.
- Characterization of isotropy orbits through rational eigenvalue properties.
- A new approach to defining submanifolds with constant principal curvatures using traceless shape operators.

## 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.