Rigidity of generalized Laplacians and some geometric applications (Q1339668)

The sheaf of \(L\)-harmonic sections is the sheaf of local solutions to \(Lu = 0\) where \(L\) is a generalized Laplacian on a manifold \(M\). The author investigates the problem of how the sheaf of \(L\)-harmonics determines \(L\) and obtains the following series of results: Two generalized Laplacians with the same sheaf of harmonics are identitical up to a multiplicative term (possibly non-constant). This property of elliptic operators is called rigidity. If the Laplacians are covariant and \(\dim M > 2\), the multiplicative term, mentioned above, is constant. This property is called strong rigidity. In particular the strong rigidity property is valid for the Laplace- Beltrami operator of any Riemannian metric on \(M\). Hence the sheaf of harmonics for \(\dim M > 2\) determines the Riemannian metric up to a multiplicative constant.