Killing tensor fields on spaces of constant curvature

A covariant symmetric tensor field ξ on a Riemannian manifold $(M, g)$ is called a Killing tensor field if the symmetrization of the covariant derivative of ξ vanishes identically.A Killing tensor field of order 1 is nothing but a Killing l-form, i .e .a l-form corresponding to a Killing vector field under the duality by means of the Riemannian metric g .The space $K(M, g)$ of all Killing tensor fields on $(M, g)$ becomes an algebra by the symmetric product.If the algebra $K(M, g)$ is generated by Killing l-forms, then the algebra of all linear differential operators on M which commutes with the Laplacian of $(M, g)$ is generated by Killing vector fields (cf.Theorem 1.1).Sumitomo-Tandai [11] proved the generation of K(Sⁿ, g) by Killing l-forms for the unit sphere Sⁿ with the standard metric $g,$ by means of the notion of pseudo-connections.This was also proved by C. Tsukamoto by representation theory of compact Lie groups.Sumitomo-Tandai [11] determined moreover the spectrum of the Lichnerowicz Laplacian Δ (Lichnerowicz [8]) on K(Sⁿ, g) , by giving explicitly projection operators of K(Sⁿ, g) onto eigenspaces of Δ .In this paper, for a two-point homogeneous space of constant curvature, we compute the dimension of the space of Killing tensor fields spanned by products of p Killing l-forms, by making use of Bott's theorem (Bott [2]) on holomorphic vector bundles over generalized flag manifolds.Together with the upper bound given by Barbance [1] for the dimension of the space KP(M, g) of Killing tensor fields of order p on a general Riemannian manifold $(M, g)$ , we prove If $(M, g)$ is a two-point homogeneous space of constant sectional curvature with M=n , then the algebra $K(M, g)$ is generated by Killing l-forms, and Kᵖ(M, g)=1/n(arrayln+p\+1array)(arrayln+p-1) , p 0 .We give furthemore an alternative determination of the spectrum of Δ on Kᵖ(Sⁿ, g) , applying the theory of spherical functions of E. Cartan to the manifold