Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Normal Holonomy of Complex Hyperbolic Submanifolds

Santiago Castañeda Montoya, Carlos E. Olmos

TL;DR

This work proves that the restricted normal holonomy group of a full Kähler submanifold of complex hyperbolic space $ abla CH^n$ is transitive when the index of relative nullity vanishes, extending Berger-type holonomy phenomena to the dual setting of complex hyperbolic geometry. The authors develop a robust framework by lifting submanifolds to the pseudo-Euclidean space $ abla^{n,1}$, introducing weakly polar actions, and handling degeneracies via holonomy tubes and the horosphere embedding. Key contributions include a precise relation between the lifted holonomy and Hermitian symmetric isotropy representations, a generalized holonomy-tube theory in degenerate contexts, and a Coxeter-group approach to force transitivity from the geometry of focal sets. Together, these techniques offer new tools for submanifold geometry in indefinite ambient spaces and shed light on the duality between complex projective and complex hyperbolic geometries.

Abstract

We prove that the restricted normal holonomy group of a Kähler submanifold of the complex hyperbolic space $\mathbb{C}H^{n}$ is always transitive, provided the index of relative nullity is zero. This contrasts with the case of $\mathbb{C}P^{n}$, where a Berger type result was proved by Console, Di Scala, and the second author. The proof is based on lifting the submanifold to the pseudo-Riemannian space $\mathbb{C}^{n,1}$ and developing new tools to handle the difficulties arising from possible degeneracies in holonomy tubes and associated distributions. In particular, we introduce the notion of weakly polar actions and a framework for dealing with degenerate submanifolds. These techniques could contribute to a broader understanding of submanifold geometry in spaces with indefinite signature, offering new insight into submanifolds in the dual setting of complex projective geometry.

Normal Holonomy of Complex Hyperbolic Submanifolds

TL;DR

This work proves that the restricted normal holonomy group of a full Kähler submanifold of complex hyperbolic space is transitive when the index of relative nullity vanishes, extending Berger-type holonomy phenomena to the dual setting of complex hyperbolic geometry. The authors develop a robust framework by lifting submanifolds to the pseudo-Euclidean space , introducing weakly polar actions, and handling degeneracies via holonomy tubes and the horosphere embedding. Key contributions include a precise relation between the lifted holonomy and Hermitian symmetric isotropy representations, a generalized holonomy-tube theory in degenerate contexts, and a Coxeter-group approach to force transitivity from the geometry of focal sets. Together, these techniques offer new tools for submanifold geometry in indefinite ambient spaces and shed light on the duality between complex projective and complex hyperbolic geometries.

Abstract

We prove that the restricted normal holonomy group of a Kähler submanifold of the complex hyperbolic space is always transitive, provided the index of relative nullity is zero. This contrasts with the case of , where a Berger type result was proved by Console, Di Scala, and the second author. The proof is based on lifting the submanifold to the pseudo-Riemannian space and developing new tools to handle the difficulties arising from possible degeneracies in holonomy tubes and associated distributions. In particular, we introduce the notion of weakly polar actions and a framework for dealing with degenerate submanifolds. These techniques could contribute to a broader understanding of submanifold geometry in spaces with indefinite signature, offering new insight into submanifolds in the dual setting of complex projective geometry.

Paper Structure

This paper contains 13 sections, 27 theorems, 103 equations.

Table of Contents

  1. Introducion
  2. Preliminaries and basic facts
  3. Riemannian isoparametric submanifolds of the Lorentz space
  4. Essentially Riemannian submanifolds
  5. Normal holonomy of essentially Riemannian submanifolds
  6. Weakly polar actions
  7. Holonomy tubes around a focal manifold
  8. The horosphere embedding
  9. The lift of a Kähler submanifold of $\mathbb CH^n$ to $\mathbb C^{n,1}$
  10. Holonomy tubes and the canonical foliation
  11. The geometry of the equivalence classes
  12. The parallel focal set
  13. The Coxeter group

Key Result

Theorem 1.1

Let $\bar{N}^n$ be a full Kähler submanifold of the complex hyperbolic space $\mathbb{C}H^{n+k}$ with zero index of relative nullity. Then the restricted normal holonomy group $\Phi$ is transitive (or equivalently, $\Phi \simeq U_k$, since it acts as a Hermitian $s$-representation).

Theorems & Definitions (60)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • ...and 50 more