## Conformally covariant equations on differential forms.(English)Zbl 0532.53021

Let M be an n-dimensional pseudo-Riemannian manifold with metric tensor g and signature (p,q) and let d and $$\delta$$ be the exterior derivative and coderivative respectively. One defines an operator $${\tilde \square}$$ on k-forms, which is a variant of the Laplacian $$\square =d\delta +\delta d$$ (more generally denoted by $$\Delta)$$, as: $$\square =((n-2k+2)/4)\delta d+((n-2k-2)/4)d\delta.$$ Further set $$m=(n-2k-2)/4$$, and denote respectively by r the Ricci tensor and the (1,1)-tensor $$\tilde r$$ defined by $$r(X,Y)=g(X,\tilde rY)$$ (X,Y any vector fields on M). Then one defines a second-order linear differential operator $${\bar \square}$$, as: ${\bar \square}A={\tilde \square}A+(4m(m+1)/(n-2))[((m+k)/(n-1))KA- \tilde r.A].$ In the above formula A is a K-form and $$\tilde r$$. is a derivation on the Grassmann algebra. The author proves some remarkable conformal quasi-invariant properties of $${\bar \square}$$ as for instance: ${\bar \square}(L_ T+m\rho)A=(L_ T+(m+1)\rho){\bar \square}A.$ In the above formula, L is the Lie derivative and T a conformal vector field on M, i.e. $$L_ T g=\rho g$$.
Reviewer: R.Rosca

### MSC:

 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 58J70 Invariance and symmetry properties for PDEs on manifolds 53A30 Conformal differential geometry (MSC2010) 58A10 Differential forms in global analysis
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