The commutative nonassociative algebra of metric curvature tensors

Abstract

It is implicit in the work of Hamilton and Huisken on the Ricci flow that the tensors of metric curvature type form a commutative nonassociative algebra for which the Killing type traceform is nondegenerate and invariant. Such an algebra is determined up to isometric isomorphism by the orthogonal group orbit of the homogeneous cubic polynomial determined by its structure tensor and the metric. The Weyl (Ricci-flat) tensors form a subalgebra, and the cubic polynomial associated to this subalgebra is harmonic, and the norm of its Hessian is a multiple of the quadratic form determined by the invariant pairing. The subalgebra of Kähler curvature tensors is also discussed and the analogous claim is proved for the subalgebra of Ricci-flat Kähler curvature tensors on a Kähler vector space. In dimension four, the self-dual Weyl tensors form a fivedimensional subalgebra equivariantly isometrically isomorphic to the deunitalization of the Jordan product on 3 × 3 symmetric matrices, and the associated cubic polynomial is the five variable cubic isoparametric polynomial of Cartan.The proofs involve the construction of explicit nontrivial idempotents in these subalgebras, and the bulk of the paper is devoted to the construction and description of idempotents. DANIEL J. F. FOX 8.2. Other invariant subalgebras of (MC W (V * ), ⊛) 79 8.3. Extending the product ⊛? 80 Acknowledgements 80 References 80