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<p>This research expository paper analyzes the interplay between symmetric spaces, holonomy groups, and the geometry of submanifolds. We begin by reviewing the Berger holonomy theorem, which classifies the possible holonomy groups of irreducible Riemannian manifolds and reveals their associated geometric structures. We then explore the role of symmetric spaces in submanifold geometry, with a focus on the normal holonomy of Euclidean submanifolds. This result was applied to homogeneous submanifolds and those with constant principal curvatures, highlighting the rigidity and symmetry of such structures. Finally, we investigate complex hyperbolic submanifolds by introducing new tools, in particular the so-called weakly polar actions, for studying pseudo-Riemannian submanifolds. This extends the techniques developed in [<CitationRef CitationID="CR7">7</CitationRef>] for obtaining a Berger-type theorem for submanifolds of complex projective space. These new results are based on [<CitationRef CitationID="CR6">6</CitationRef>].We hope this overview might provide useful perspectives on these classical yet evolving topics.</p>

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Submanifolds and holonomy in pseudo-Riemannian space forms

  • 期刊论文

  • 发表日期：

    2026年7月20日

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  • Santiago Castañeda-Montoya,
  • Carlos E. Olmos

摘要

This research expository paper analyzes the interplay between symmetric spaces, holonomy groups, and the geometry of submanifolds. We begin by reviewing the Berger holonomy theorem, which classifies the possible holonomy groups of irreducible Riemannian manifolds and reveals their associated geometric structures. We then explore the role of symmetric spaces in submanifold geometry, with a focus on the normal holonomy of Euclidean submanifolds. This result was applied to homogeneous submanifolds and those with constant principal curvatures, highlighting the rigidity and symmetry of such structures. Finally, we investigate complex hyperbolic submanifolds by introducing new tools, in particular the so-called weakly polar actions, for studying pseudo-Riemannian submanifolds. This extends the techniques developed in [7] for obtaining a Berger-type theorem for submanifolds of complex projective space. These new results are based on [6].We hope this overview might provide useful perspectives on these classical yet evolving topics.