Helgason spheres of compact symmetric spaces and immersions of finite type (Q2721593)

Let \((M,g)\) be a compact irreducible symmetric space. A Helgason sphere in~\((M,g)\) is a maximal totally geodesic submanifold with maximal sectional curvature. A Helgason circle is a circle in a Helgason sphere. In this article, the author shows that every Helgason circle is a homogeneous curve and that any two Helgason circles with the same radius are conjugate. Next, he links Helgason spheres and Helgason circles with the theory of immersions of finite type in Euclidean spaces. Recall that an immersion \(i: M\to E^m\) is of finite type if every coordinate function \(x_k\circ i: M\to {\mathbb R}\) is a (finite) linear combination of eigenfunctions of the Laplacian on~\((M,g)\). In particular, he proves that an isometric immersion of class \(C^\omega\) is of finite type if and only if it carries each Helgason circle, resp.\ each Helgason sphere, into a curve of finite type, resp.\ a submanifold of finite type.