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The Minimal Polynomial of a Riemannian C_0-Space

Tillmann Jentsch

TL;DR

This work introduces a curvature-based invariant for Riemannian $\mathfrak{C}_0$-spaces by constructing a pointwise minimal polynomial $P_{\min}(M,g,p)$ from the symmetrized covariant derivatives of the curvature. On homogeneous spaces, these pointwise polynomials glue to a global polynomial with coefficients that are Killing tensors invariant under the full isometry group, and the polynomial degree provides an upper bound for the Singer invariant. The paper develops an algebraic framework ensuring existence and uniqueness of such polynomials under a kernel-ideal condition, and proves that for $\mathfrak{C}_0$-spaces the coefficients become global invariant tensors on suitable regions; it also provides explicit examples and discusses the implications for geometric structure and symmetry. The results connect deep algebraic properties of curvature jets with tangible geometric objects, offering both concrete invariants and a path to understanding symmetry in Riemannian geometry. Overall, the minimal polynomial serves as a bridge between curvature derivatives, Killing tensors, and invariants of symmetry in C0-spaces, with potential to illuminate homogeneity and stability phenomena in geometric analysis.

Abstract

We construct, at every point of a Riemannian C_0-space, a polynomial in one variable whose coefficients are polynomial functions on the tangent space. For a homogeneous Riemannian C_0-space (for instance, a G.O. space) these pointwise-defined polynomials glue to a global polynomial whose coefficients are Killing tensors invariant under the full isometry group. Moreover, the degree of this polynomial provides an upper bound for the Singer invariant of the space.

The Minimal Polynomial of a Riemannian C_0-Space

TL;DR

This work introduces a curvature-based invariant for Riemannian -spaces by constructing a pointwise minimal polynomial from the symmetrized covariant derivatives of the curvature. On homogeneous spaces, these pointwise polynomials glue to a global polynomial with coefficients that are Killing tensors invariant under the full isometry group, and the polynomial degree provides an upper bound for the Singer invariant. The paper develops an algebraic framework ensuring existence and uniqueness of such polynomials under a kernel-ideal condition, and proves that for -spaces the coefficients become global invariant tensors on suitable regions; it also provides explicit examples and discusses the implications for geometric structure and symmetry. The results connect deep algebraic properties of curvature jets with tangible geometric objects, offering both concrete invariants and a path to understanding symmetry in Riemannian geometry. Overall, the minimal polynomial serves as a bridge between curvature derivatives, Killing tensors, and invariants of symmetry in C0-spaces, with potential to illuminate homogeneity and stability phenomena in geometric analysis.

Abstract

We construct, at every point of a Riemannian C_0-space, a polynomial in one variable whose coefficients are polynomial functions on the tangent space. For a homogeneous Riemannian C_0-space (for instance, a G.O. space) these pointwise-defined polynomials glue to a global polynomial whose coefficients are Killing tensors invariant under the full isometry group. Moreover, the degree of this polynomial provides an upper bound for the Singer invariant of the space.
Paper Structure (19 sections, 20 theorems, 151 equations)

This paper contains 19 sections, 20 theorems, 151 equations.

Table of Contents

  1. Overview
  2. The minimal polynomial of a Riemannian C0-space
  3. Pointwise properties of the minimal polynomial of a Riemannian $\mathfrak{C}_0$-space
  4. Global results
  5. Examples
  6. Estimates for the Singer invariant of a $\mathfrak{C}_0$-space
  7. Concluding remarks
  8. Existence of the minimal polynomial
  9. An ideal-theoretic criterion for polynomial minimality
  10. The minimal polynomial of a C0-space
  11. Proof of Theorem \ref{['th:main']}
  12. Pointwise minimal polynomials of a C0-space
  13. Proofs of Theorem \ref{['th:main_addition']} and Corollary \ref{['co:main_addition']}
  14. Proof of Theorem \ref{['th:main_global']}
  15. Proof of Theorem \ref{['th:main_global']}
  16. ...and 4 more sections

Key Result

Theorem 1.6

Let $(M,g)$ be a $\mathfrak{C}_0$-space. Then the minimal polynomial $P_{\mathrm{min}}(M,g,p)$ exists at every $p \in M$ and generates the kernel of the evaluation map eq:eval_2 as an ideal in $\mathscr{P}(T_pM)[\lambda]$.

Theorems & Definitions (42)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5: BTV
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 32 more