Attached Submanifolds Beyond Symmetric Spaces

TL;DR

This paper extends the concept of attached submanifolds from symmetric spaces to more general solvmanifolds, establishing conditions under which Ricci curvature restrictions hold and providing new examples beyond symmetric spaces.

Contribution

It generalizes Tamaru's construction of attached submanifolds by weakening assumptions, introduces the Jacobi Star Condition, and demonstrates the existence of attached submanifolds in non-symmetric solvmanifolds.

Findings

Ricci curvature restriction property holds under the Jacobi Star Condition

Attached submanifolds are minimal and sometimes totally geodesic

Existence of attached submanifolds in non-symmetric solvmanifolds

Abstract

We study submanifold geometry in the presence of symmetry, focusing on submanifolds of solvmanifolds with an unusual property relative to Ricci curvature. We generalize work of H. Tamaru \cite{tamaru-11} in which he explores the geometry of submanifolds of symmetric spaces of noncompact type constructed from parabolic subgroups of the isometry group. He calls these attached submanifolds. The Ricci curvatures of attached submanifolds coincide with the restrictions of the Ricci curvatures of ambient symmetric spaces. We broaden Tamaru's construction by weakening the hypotheses on the ambient space, allowing a pseudo-Riemannian scalar product, and defining attached submanifolds in terms of root spaces. We demonstrate that in this setting, the Ricci curvature restriction property for attached submanifolds holds if and only if the submanifold satisfies an algebraic criterion that we call the Jacobi Star Condition. Like attached submanifolds of symmetric spaces, our attached submanifolds are minimal, and are only totally geodesic under hypotheses analogous to hypotheses in the symmetric space case. Finally, we give an example of a solvmanifold that has an attached submanifold and is not a symmetric space, demonstrating that attached submanifolds are not unique to symmetric spaces.