Non-elliptic Structure of the Ricci Operator

Rjk = R̄jk + ∇̄mD jk − ∇̄jD mk −Dn jkD nm −Dn mkD nj . (To see why this is true, just compute in local normal coordinates for ḡ!) From now on all components are tensorial (as opposed to local coordinate components.) The terms involving second covariant derivatives of the metric are: 1 2 g(−∇̄m∇̄lgjk +∇̄m∇̄kgjl +∇̄m∇̄jglk +∇̄j∇̄lgmk−∇̄j∇̄kgml−∇̄j∇̄mglk). The first term is a kind of Laplacian. The third and sixth terms add up to a commutator of covariant derivatives, a lower-order (curvature) termso we drop them. To make the expression formally symmetric in j, k, we change the order of ∇̄l and ∇̄j in the fourth term (again at the cost of a lower-order term). With these changes, to identify the remaining terms (starting at the second) we write (commuting covariant derivatives when needed):