Prolongation and Killing two-tensors

Abstract: We present a systematic prolongation procedure and its implementation for Killing two-tensors, especially in the locally symmetric case. We use the resulting machinery to elucidate the natural quadratic mapping from Killing fields to Killing two-tensors on irreducible locally symmetric spaces of compact type.

• The paper introduces a two-step prolongation scheme that converts the Killing two-tensors equation into a first-order connection with computable curvature.
• It provides explicit algebraic criteria distinguishing decomposable tensors from hidden symmetries using representation-theoretic techniques and computational tools like LiE.
• The framework yields dimensionally sharp results on symmetric spaces and paves the way for computational implementations in symmetry analysis and integrable systems.

Systematic Prolongation and the Structure of Killing Two-Tensors

Introduction and Motivation

The paper "Prolongation and Killing two-tensors" (2604.17855) develops a comprehensive prolongation framework for Killing two-tensors, with special emphasis on the locally symmetric case. It establishes a canonical, two-step prolongation scheme that provides a direct correspondence between Killing two-tensors and parallel sections of an explicitly constructed vector bundle with a compatible connection. The approach not only recovers existing classification results in constant curvature cases but also gives explicit algebraic criteria for the decomposition of Killing two-tensors and the presence of so-called "hidden symmetries" on irreducible locally symmetric spaces of compact type. The article furthermore addresses the injectivity of the natural quadratic map from Killing fields to Killing two-tensors, exploiting the fact that the algebraic theory of highest weight representations and computational tools like LiE yield concrete answers in diverse geometric settings.

Prolongation Theory

The authors detail a systematic prolongation construction for overdetermined linear systems defined by torsion-free connections. The general framework takes a vector bundle $U$ with a connection and a subbundle $V$ realized via a differential operator, and iteratively constructs prolongation bundles and connections encoding the solution spaces of the original PDEs as parallel sections.

The central result is a canonical, functorial procedure (unique up to automorphisms induced by the ambiguity in the choice of curvature lifts) giving a prolonged connection whose parallel sections correspond bijectively to solutions of the original system. In the context of Killing fields (i.e., vector fields with vanishing symmetrized covariant derivative) and Killing two-tensors (symmetric tensors killed by the symmetrized covariant derivative), this prolongation turns a strongly overdetermined system into a sequence of bundles of increasing rank but decreasing order, culminating in an explicit first-order connection with computable curvature.

Killing Fields and Killing Two-Tensors

For Killing one-forms, the prolongation is classical, producing a rank-$(n + n(n-1)/2)$ bundle whose parallel sections are in bijection with the space of Killing fields. The curvature of the prolonged connection is explicitly computed and carries tight algebraic constraints stemming from the Bianchi identity and Riemann tensor symmetries.

For Killing two-tensors, the situation is more intricate; the PDE is even more overdetermined, and further prolongation is required. The article constructs a sequence:

1. The first prolongation encodes Killing two-tensors and associated lower order data into a bundle, but there remains ambiguity up to a certain "hook" Young symmetry type, reflecting the nonuniqueness of the curvature lift.
2. The second prolongation removes this ambiguity and, in locally symmetric spaces, results in a canonical identification of Killing two-tensors as parallel sections of a bundle, where the Riemann curvature tensor and its symmetries govern the allowed forms of the connection and its curvature.

The principal connection formula for Killing two-tensors in the locally symmetric (i.e., $\nabla_a R_{bc}{}^d{}_e = 0$) setting is made explicit. The curvature of the resulting bundle is traced back to the algebraic structure of the ambient space, with the bundle splitting controlled by representation-theoretic data.

Algebraic Decomposition and Hidden Symmetries

An essential contribution of the paper is the clarification of the structure of decomposable vs. hidden Killing two-tensors. Decomposable tensors arise as symmetric products of Killing fields; hidden symmetries are those Killing two-tensors not directly generated via this quadratic product.

A key insight is that the kernel of the quadratic map from Killing fields to two-tensors is controlled by Killing-Yano three-forms, with a systematic algebraic short exact sequence relating the involved bundles:

$$0 \to \left(\Lambda^3 \oplus \Lambda^4\right) \to \odot^2(\Lambda^1 \oplus \Lambda^2) \to P \to 0,$$

with precise geometric meaning for each term. The presence or absence of hidden symmetries is then encoded in the vanishing or nonvanishing of certain representations (in particular, Killing-Yano forms) within the cohomology of these sequences as modules over the local holonomy algebra.

For compact type, irreducible symmetric spaces, the quadratic mapping is injective except for spheres and compact Lie groups with their bi-invariant metrics, where explicit "extra" symmetries (e.g., the Cartan 3-form) provide nontrivial kernel elements. This dichotomy is described with precise representation-theoretic language using the full machinery of highest weights.

Numerical Results and Representation-Theoretic Analysis

The authors provide explicit, dimensionally sharp statements for the spaces of Killing two-tensors across families of symmetric spaces. For example, on the sphere $S^n$, the dimension of the space of Killing two-tensors is $n(n+1)^2(n+2)/12$; on the complex projective space $\mathbb{CP}_m$ it is $m(m+1)^2(m+2)/2$; on quaternionic and octonionic projective spaces these numbers are computed using recent advances [MN].

For specific spaces such as $E_6/F_4$, the authors confirm (via branching rules and bundled representation theory) that the dimension of the Killing two-tensors strictly exceeds the dimension predicted by decomposables, providing an explicit irreducible $V$0-representation of hidden symmetries. These calculations are efficiently tractable with the aid of LiE and confirm and extend recent results in the field.

Implications and Future Directions

Practically, the presented prolongation machinery allows for algorithmic computation of the space of Killing two-tensors, making the method amenable for symbolic and computational implementation on a variety of symmetric spaces. The theoretical implications are significant: the explicit algebraic exact sequences, representation-theoretic decompositions, and connection formulas provide a comprehensive classification scheme for second-order Killing tensors, unlocking further study of geodesic flows, conserved quantities in integrable systems, and symmetry analysis in general relativity.

The adaptability of the framework to the affine case (without metric) is briefly discussed, hinting at generalizations to other parabolic geometries where similar prolongation techniques and sequences might apply. Additionally, the explicit machinery developed for handling the kernel of the quadratic map suggests further study of higher-order Killing tensor fields and their relationship to the underlying holonomy and curvature invariants.

Conclusion

This work advances the analytic and algebraic theory of Killing two-tensors by providing a canonical prolongation scheme and a representation-theoretic classification of their structure on locally symmetric spaces. The method allows for explicit computation; clarifies the geometric and algebraic nature of hidden symmetries; and paves the way for computational and theoretical advancements in the study of geometric PDEs and the algebraic structure of symmetries in differential geometry (2604.17855).