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

Branson, Thomas P.

Commun. Partial Differ. Equations 7, 393-431 (1982).

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

Keywords:

conformal quasi-invariance; pseudo-Riemannian manifold; Laplacian

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