[Paper Review] On a new class of infinitesimal group actions on pseudo-Riemannian manifolds

Sigbjørn Hervik|arXiv (Cornell University)|May 23, 2018

Geometric Analysis and Curvature Flows1 references3 citations

TL;DR

This paper introduces Nil-Killing vector fields—vector fields whose Lie derivative of the metric is nilpotent—as a new geometric structure that explains why certain pseudo-Riemannian manifolds are $χ$-degenerate (i.e., have identical polynomial curvature invariants). By generalizing Killing vectors, these fields generate one-parameter groups of diffeomorphisms preserving all scalar curvature invariants, thereby providing a local mechanism for CSI (constant scalar curvature invariants) spacetimes. The key contribution is that all known $χ$-degenerate examples possess such Nil-Killing Lie algebras, offering a unified explanation for their CSI property beyond local homogeneity.

ABSTRACT

Using the Lie derivative of the metric we define a class of Lie algebras of vector fields by generalising the concept of Killing vectors. As a Lie algebra they define locally a group action on the pseudo-Riemannian manifold through exponentiation. The motivation behind studying these infinitesimal group actions is the investigation of $\mathcal{I}$-degenerate pseudo-Riemannian spaces, i.e., spaces having identical polynomial curvature invariants. In particular, we show that all the known examples of $\mathcal{I}$-degenerate pseudo-Riemannian spaces possess such vector fields.

Motivation & Objective

• To explain the CSI (constant scalar curvature invariants) property in pseudo-Riemannian manifolds that are not locally homogeneous, particularly in neutral signature spacetimes.
• To identify a new class of vector fields—Nil-Killing fields—that preserve all polynomial curvature invariants, generalizing Killing vectors.
• To establish a local geometric mechanism (via Lie algebras of Nil-Killing fields) that ensures the invariance of curvature invariants under diffeomorphisms.
• To provide a unifying framework for understanding known $χ$-degenerate metrics, including those in four-dimensional neutral signature that lack sufficient Killing vectors.

Proposed method

• Define an $χ$-preserving diffeomorphism (IPD) as a diffeomorphism that leaves all scalar polynomial curvature invariants unchanged.
• Introduce Nil-Killing vector fields via the condition that the Lie derivative of the metric is a nilpotent endomorphism on the tangent bundle.
• Construct the Nil-Killing Lie algebra as a finite-dimensional Lie algebra of vector fields whose metric Lie derivatives are nilpotent operators.
• Show that such vector fields generate one-parameter groups of IPDs, implying invariance of curvature invariants under their flow.
• Demonstrate that known $χ$-degenerate metrics (e.g., in neutral signature) admit abelian or solvable Nil-Killing Lie algebras spanned by coordinate vector fields like $\partial_{v_i}$.
• Use coordinate transformations and limits to prove that the nilpotency of the metric Lie derivative holds at all points, ensuring global nilpotency of the associated operator.

Experimental results

Research questions

• RQ1 Under what conditions does the set of Nil-Killing vector fields form a Lie algebra?
• RQ2 Does a generic pseudo-Riemannian space admit at least one non-Killing Nil-Killing vector field?
• RQ3 Does the existence of a non-Killing Nil-Killing vector field imply that the spacetime is $χ$-degenerate?
• RQ4 Given two $χ$-degenerate metrics with identical curvature invariants, do they necessarily have isomorphic maximal Nil-Killing Lie algebras?
• RQ5 Under what conditions do Nil-Killing vector fields generate $χ$-preserving diffeomorphisms (IPDs)?

Key findings

• All known examples of $χ$-degenerate pseudo-Riemannian manifolds, including four-dimensional neutral signature metrics with only three Killing vectors, possess a transitive set of Nil-Killing vector fields.
• The metric $ds^2 = 2du(dv + V du) + 2dU(dV + b v^4 dU)$ is shown to be CSI due to the existence of a five-dimensional solvable Nil-Killing Lie algebra, including $\partial_u, \partial_v, \partial_U, \partial_V$, and $\xi_5 = -u\partial_u + v\partial_v - 2U\partial_U + 2V\partial_V$.
• The Lie derivative of the metric along each Nil-Killing vector is a nilpotent endomorphism, ensuring that all polynomial invariants of the metric derivative vanish, which implies invariance of curvature invariants under the flow.
• The vector fields $\partial_{v_i}$ in the general class of $χ$-degenerate metrics generate translations that preserve all curvature invariants, forming an abelian Nil-Killing Lie algebra of dimension equal to the real rank of $O(p,q)$.
• The limit $\lim_{t\to\infty} \phi_t^* n_{ab} = 0$ confirms that the operator $n_{ab}$ is nilpotent at every point, validating the nilpotency condition globally.
• The construction of the general $χ$-degenerate metric class (12) shows that $\mathcal{N} = \mathrm{span}\{\partial_{v_1}, \dots, \partial_{v_k}\}$ is a Nil-Killing Lie algebra, providing a systematic way to generate such spaces.