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Inhomogeneous almost symmetric submanifolds

Claudio Gorodski, Carlos Olmos

TL;DR

This work completes the classification of inhomogeneous almost symmetric submanifolds of Euclidean space by reducing to irreducible $s$-representations and a controlling 1D base curve $L$, from which full irreducible submanifolds $KL$, $(SO(k) imes K)L$, and $(SO(k) imes SO(k') imes K)L$ are constructed. The approach shows every such submanifold arises from this construction, and a holonomy-based argument demonstrates these examples are inhomogeneous unless they reduce to a round sphere. The results bridge extrinsic symmetry, representation theory, and holonomy, and the appendix establishes a uniqueness result for extrinsic splittings of isometric immersions. Overall, the paper provides a complete, rigidity-driven description of inhomogeneous almost symmetric submanifolds and clarifies when symmetry arises from ambient group actions versus intrinsic curvature structure.

Abstract

We completely describe inhomogeneous properly embedded almost symmetric submanifolds of Euclidean space as certain unions of parallel symmetric submanifolds of the ambient Euclidean space.

Inhomogeneous almost symmetric submanifolds

TL;DR

This work completes the classification of inhomogeneous almost symmetric submanifolds of Euclidean space by reducing to irreducible -representations and a controlling 1D base curve , from which full irreducible submanifolds , , and are constructed. The approach shows every such submanifold arises from this construction, and a holonomy-based argument demonstrates these examples are inhomogeneous unless they reduce to a round sphere. The results bridge extrinsic symmetry, representation theory, and holonomy, and the appendix establishes a uniqueness result for extrinsic splittings of isometric immersions. Overall, the paper provides a complete, rigidity-driven description of inhomogeneous almost symmetric submanifolds and clarifies when symmetry arises from ambient group actions versus intrinsic curvature structure.

Abstract

We completely describe inhomogeneous properly embedded almost symmetric submanifolds of Euclidean space as certain unions of parallel symmetric submanifolds of the ambient Euclidean space.
Paper Structure (6 sections, 12 theorems, 27 equations)

This paper contains 6 sections, 12 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminary material
  3. Characterization of inhomogeneous almost symmetric submanifolds
  4. Construction of inhomogeneous almost symmetric submanifolds
  5. Inhomogeneity
  6. Appendix: Uniqueness of extrinsic splittings of isometric immersions

Key Result

Theorem 1.1

Consider an irreducible s-representation of a compact connected Lie group $K_i$ on $V_i$ with a fixed non-trivial extrinsic symmetric orbit $K_i\cdot v_i$, for $i=1, \ldots , r$. Put $V = V_0\oplus V_1\oplus \cdots \oplus V_r$, where $V_0$ is a possibly trivial Euclidean space, and consider the diag Further, the submanifolds constructed above are inhomogeneous, unless they are a round sphere. Conv

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Lemma 4.2
  • ...and 3 more