Spektrale Geometrie und Huygenssches Prinzip für Tensorfelder und Differentialformen I

Abstract

Several Laplace operators $L$ over an $n$-dimensional Riemannian manifold ($M,g$), applying to tensor fields and (alternating or symmetric) differential forms of degree $p$, are considered. For definite $g$ and closed $M$ results on spectral geometry are obtained. If e.g. some $L$ is iso-spectral to a Laplacian of a flat manifold and $p$ greater than some bound depending on $n$, then $(M,g)$ is flat. For lorentzian $g$ and $n=6$ results on Huygens’ principle (HP) are obtained. From $HP$ and additional assumptions there follows that $(M,g)$ is flat.