On Killing tensors on Riemannian symmetric spaces

Abstract: A Killing tensor field on a Riemannian space corresponds to an integral of the geodesic flow polynomial in momenta. A Killing tensor field is called decomposable if it is a polynomial in Killing vector fields. In this paper, we first prove that the study of Killing tensor fields on symmetric spaces can be reduced to the case of compact irreducible ones. Then we introduce the class of top slot Killing tensor fields. We obtain an explicit and elegant description of such tensor fields and prove that the quadratic Killing tensor fields are spanned by the top-slot ones. We also show that quadratic Killing tensor fields on the quaternionic projective space and on the Cayley projective space are spanned by the indecomposable ones constructed in our earlier paper and the decomposable ones. This completes the classification of quadratic Killing tensor fields on Riemannian symmetric spaces of rank one.

• The paper presents a reduction of the classification problem to compact irreducible symmetric spaces via De Rham decomposition and duality.
• It introduces explicit linear criteria for quadratic Killing tensors using the top slot concept, enabling computer-aided approaches.
• The work resolves the classification in rank-one symmetric spaces by distinguishing decomposable and indecomposable Killing tensors.

Killing Tensors on Riemannian Symmetric Spaces: Structural and Classification Results

Introduction and Context

The paper "On Killing tensors on Riemannian symmetric spaces" (2604.27950) provides a comprehensive structural and classification analysis of Killing tensor fields in the setting of Riemannian symmetric spaces. A Killing tensor field of rank $d$ is a symmetric tensor $L$ satisfying the Killing equation $L_{(i_1 \dots i_d, j)} = 0$, encoding polynomial integrals for the geodesic flow. Killing tensors of rank $1$ correspond to Killing vector fields (infinitesimal isometries), while higher rank cases encode more complex integrable structures. Decomposable tensors are symmetric polynomials in Killing vector fields; indecomposable tensors are not expressible as such and are key to integrability phenomena and the Mishchenko–Fomenko problem.

The structure of the algebra of Killing tensor fields is fully understood for spaces of constant curvature, but symmetric spaces (beyond constant curvature) present rich and challenging algebraic and geometric structures with potential for both decomposable and indecomposable Killing tensors.

Algebraic Reductions and Duality

A central contribution is the reduction of the general classification problem for Killing tensors on symmetric spaces to the study of compact irreducible symmetric spaces. This is achieved via two structural theorems:

• De Rham Decomposition (Theorem 1): For a symmetric space $(M,g)$ with its de Rham decomposition $(M_0, g_0) \times \dots \times (M_m, g_m)$, the algebra of Killing tensors on $M$ is the tensor product of the algebras on the factors. This result fails for generic Riemannian products unless imposed by symmetric space structure, but holds in the symmetric space setting due to vanishing covariant derivative terms for curvature.
• Duality (Theorem 2): The complexified graded associative algebras of Killing tensors on a pair of dual globally symmetric spaces (one compact, one noncompact type) are isomorphic. It follows that classification in one case suffices for the other, modulo explicit sign transformations in curvature and tensor expansions.

Thus, for the remainder of the analysis, attention is focused on compact irreducible symmetric spaces.

Quadratic Killing Tensors: Explicit Classification and Top Slot Structure

A major technical achievement is the explicit classification of quadratic Killing tensor fields, facilitated by the introduction of the "top slot" concept:

• Top Slot Killing Tensors: A tensor is top slot at a point if its $(d-1)$-st jet vanishes; for quadratic tensors ($d=2$), this means the Taylor expansion is determined by a single constant tensor $K_2$ of type $L$0, symmetric in first and last pairs of arguments.
• Classification Theorem (Theorem 3): For any compact irreducible symmetric space, the space of quadratic Killing tensors is spanned by top slot tensors (i.e., those whose Taylor expansion at a point is determined solely by $L$1). Necessary and sufficient linear algebraic conditions are imposed on $L$2, depending only on the curvature tensor. This includes:
  • Vanishing cyclic sums: $L$3.
  • Compatibility relations with the curvature: $L$4, and analogous higher-order conditions.

These structural results are extended to arbitrary rank for top slot tensors (Proposition: conditions for Taylor expansion and integrability generalize), with explicit linear conditions analogous to those in quadratic case. However, whether all Killing tensors of higher rank are spanned by top slot ones remains open.

Rank-One Symmetric Spaces: Complete Classification

The paper completes the classification of quadratic Killing tensors in rank-one symmetric spaces (spheres, complex projective spaces, quaternionic projective spaces, Cayley projective planes):

• Spheres and Complex Projective Spaces: All Killing tensors are decomposable; this follows from prior literature and is reproven geometrically.
• Quaternionic Projective Spaces $L$5 ($L$6): There exists a family of indecomposable quadratic Killing tensors not expressible as symmetric products of Killing vector fields, constructed explicitly by exploiting the quaternionic structure and associated symmetric matrices. The space of Killing tensors is spanned by these indecomposable elements and decomposables.
• Cayley Projective Plane $L$7: Similar indecomposable quadratic Killing tensors exist, constructed using the exceptional Lie group $L$8 and the structure of the Albert algebra.
• For $L$9: All quadratic Killing tensors are decomposable.

This classification closes the longstanding question for rank-one spaces. Furthermore, computationally strong results are presented, including explicit dimension counts for spaces of tensors with the required symmetries and demonstration that certain invariant tensors (e.g., $L_{(i_1 \dots i_d, j)} = 0$0) are not decomposable in the sense of symmetric space representation theory.

Implications and Future Directions

The structural reductions (via De Rham and duality theorems) enable streamlined classification strategies for Killing tensors in symmetric spaces, suggesting central roles for representation-theoretic and computational methods. The explicit linear criteria for top slot tensors facilitate algorithms for further classification and open the possibility for computer-aided exploration in higher rank settings.

Indecomposable Killing tensors, as exhibited in quaternionic and Cayley projective spaces, have practical implications for integrable systems on homogeneous spaces, particularly in constructing Poisson-commutative subalgebras with low-degree integrals—directly relevant to the Mishchenko–Fomenko method and its quantum generalizations.

The open question of whether top slot tensors span all Killing tensors in higher rank remains central, with resolution likely connected to deep representation theory and algebraic geometry of symmetric spaces. There are implications for integrability, symmetry reduction in PDEs, and geometric quantization.

Conclusion

This work establishes a precise structural framework and classification for quadratic Killing tensors on Riemannian symmetric spaces, resolves fundamental questions for rank-one cases, and provides explicit algebraic and analytic tools for further exploration. The reductions to compact irreducible cases, duality correspondence, and explicit linear criteria for top slot tensors represent significant advances in the geometric theory of integrability and the algebra of tensor invariants. The practical and theoretical ramifications are broad, impacting integrable systems, geometric analysis, and representation theory in Riemannian geometry and mathematical physics.