The geometry and the Laplace operator on the exterior $2$-forms on a compact Riemannian manifold

Abstract:

A compact, orientable, Riemannian manifold of dimension n is considered, with the Laplace operator acting on the exterior 2-forms of the manifold. Examining the spectrum, ${\text {Sp}^2}$, of the Laplace operator acting on 2-forms, the question is raised whether ${\text {Sp}^2}$ exerts an influence on the geometry of the Riemannian manifold. To answer this question, after some preliminaries, two compact, orientable, equispectral, i.e., having the same ${\text {Sp}^2}$, Riemannian manifolds are considered in §3. (We note, in particular, that equispectral implies that the two manifolds are equidimensional.) Assuming further that the second Riemannian manifold has constant sectional curvature, the paper exhibits all the dimensions, commencing with 2, for which the two Riemannian equispectral manifolds have the same constant sectional curvature. In particular, this implies that for certain dimensions, which are explicitly stated, the Euclidean n-sphere is completely characterized by the spectrum, ${\text {Sp}^2}$, of the Laplacian on exterior 2-forms. Next, two compact, orientable, equispectral, Einsteinian manifolds are considered. (Again, equispectral implies equidimensional.) Assuming that the second Einsteinian manifold is of constant sectional curvature, the paper exhibits all the dimensions for which the two Einsteinian equispectral manifolds have equal constant sectional curvature. In particular, taking the second manifold to be the standard Euclidean sphere, the paper classifies Einsteinian manifolds, which are equispectral to the sphere, by calculating all the dimensions for which the Einsteinian manifold is isometric to the sphere. In short, if one of the Einsteinian manifolds is the sphere, then for certain dimensions, equispectral implies isometric. In §4, compact, equispectral, Kählerian manifolds are considered, and additional conditions are examined which determine their geometry. Studying two compact, equispectral, Kählerian manifolds, and again assuming that one of the manifolds is of real, constant, holomorphic, sectional curvature, the paper exhibits all the dimensions for which the two manifolds have equal real, constant, holomorphic, sectional curvatures. As a particular case, the paper classifies all the dimensions for which complex projective space, with Fubini-Study metric, is completely characterized by the spectrum, ${\text {Sp}^2}$, of the Laplacian acting on exterior 2-forms. The calculations were performed by utilizing an electronic computer.