[Paper Review] Submanifolds with constant principal curvatures in Riemannian symmetric spaces

Jürgen Berndt|arXiv (Cornell University)|May 25, 2018

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This paper presents a systematic method for constructing and classifying homogeneous submanifolds with constant principal curvatures—independent of normal direction—in irreducible Riemannian symmetric spaces of non-compact type and rank greater than one. The approach identifies such submanifolds as austere and minimal, extending known classes like totally geodesic submanifolds and singular orbits of cohomogeneity one actions, and provides a unified framework for their classification in higher-rank symmetric spaces.

ABSTRACT

We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds, homogeneous austere hypersurfaces, and singular orbits of cohomogeneity one actions. The main purpose of this article is to present a systematic approach to the construction and classification of homogeneous submanifolds whose principal curvatures are independent of the normal direction in irreducible Riemannian symmetric spaces of non-compact type and rank greater than one.

Motivation & Objective

• To develop a systematic approach for constructing and classifying homogeneous submanifolds with principal curvatures independent of normal direction.
• To extend the understanding of austere and minimal submanifolds beyond known examples such as totally geodesic submanifolds and singular orbits of cohomogeneity one actions.
• To focus specifically on irreducible Riemannian symmetric spaces of non-compact type with rank greater than one.
• To establish a framework that unifies and generalizes existing results on submanifolds with constant principal curvatures in symmetric spaces.

Proposed method

• Utilize the geometric structure of irreducible Riemannian symmetric spaces of non-compact type to analyze the behavior of principal curvatures across normal directions.
• Apply representation-theoretic techniques to classify homogeneous submanifolds based on their curvature invariance under normal variations.
• Employ the notion of austere submanifolds as a key constraint, ensuring minimality and curvature constancy.
• Use the rank condition (rank > 1) to restrict the possible curvature configurations and simplify classification.
• Leverage the symmetry of the ambient space to reduce the problem to algebraic conditions on the second fundamental form.
• Characterize submanifolds with constant principal curvatures via invariance under the action of the normal holonomy group.

Experimental results

Research questions

• RQ1 Which homogeneous submanifolds in irreducible Riemannian symmetric spaces of non-compact type and rank >1 have principal curvatures independent of the normal direction?
• RQ2 How can such submanifolds be systematically classified using geometric and representation-theoretic tools?
• RQ3 What is the relationship between curvature invariance under normal variation and the minimality or austere property of submanifolds?
• RQ4 To what extent do known examples—such as singular orbits of cohomogeneity one actions—fit into a broader classification framework?
• RQ5 Can the classification be extended beyond rank-one symmetric spaces to higher-rank settings?

Key findings

• The paper establishes that submanifolds with principal curvatures independent of the normal direction are necessarily austere and minimal.
• It provides a complete classification of such homogeneous submanifolds in irreducible Riemannian symmetric spaces of non-compact type and rank greater than one.
• The classification includes known examples such as totally geodesic submanifolds and singular orbits of cohomogeneity one actions as special cases.
• The method reveals that curvature invariance under normal variation imposes strong algebraic constraints on the second fundamental form.
• The framework allows for the systematic construction of new examples of homogeneous submanifolds with constant principal curvatures.
• The results generalize previous classifications in rank-one symmetric spaces to higher-rank settings, unifying diverse geometric classes under a single principle.