Švec, Alois Vector fields on hyperspaces. (English) Zbl 0634.58037 Czech. Math. J. 37(112), 207-230 (1987). Let (M,g) be a Riemannian manifold. We denote by \(\nabla\) the associated linear connection. We define the Laplace operator \(\Delta_ 0\) for a q- covariant tensor T and another q-covariant tensor \(\Delta_ 0T\) which in a local coordinate system \((x^ 1,...x^ n)\) has the form \((\Delta_ 0T)_{i...j}=g^{k\ell}\nabla_ k\nabla_{\ell}T_{i...j}\). For \(g=0\), \(\Delta_ 0\) coincides with the classical Laplacian \(\Delta\) on functions; for \(q>0\) and T skew-symmetric, \(\Delta_ 0\) does not differ too much from the classical Laplacian \(\Delta =-(d\delta +\delta d)\) on q-forms on M. From this operator \(\Delta_ 0\) we obtain \(Sp_{[q]}(M)\{\lambda \in {\mathbb{R}}|\Delta_ 0T=-\lambda T\}\). Because of the existence of the metric, we may study contravariant tensors as well. The Laplacian of the tensor \(\{T_{j_ 1...j_ s}^{i_ 1...i_ r}\}\) is defined by \[ (\Delta_ 0T)^{i_ 1...i_ r}_{j_ 1...j_ s}\equiv \Delta_ 0T^{i_ 1...i_ r}_{j_ 1...j_ s}=g^{ij}T^{i_ 1...i_ r}_{j_ 1...j_ s,ij}. \] We define \(Spec_{(r,s)}(M,ds^ 2)=\{\lambda \in {\mathbb{R}}|\Delta_ 0T_{j_ 1...j_ s}^{i_ 1...i_ r}=-\lambda T_{j_ 1...j_ s}^{i_ 1...i_ r}\}\). This tensor \(T=\{T_{j_ 1...j_ 1}^{i_ 1...i_ r}\}\) is called an eigentensor. These form an \({\mathbb{R}}\)-module denoted by \(E^{\lambda}_{(r,s)}(M,ds^ 2)\). The aim of the present paper is to study \(Spec_{(r,s)}(M,ds^ 2)\), \(E^{\lambda}_{(r,s)}(M,ds^ 2)\) and \(E_{(r,s)}(M,ds^ 2)\). Reviewer: G.Tsagas Cited in 2 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C20 Global Riemannian geometry, including pinching Keywords:spectrum; vector field; tensor field; Riemannian manifold; Laplace operator PDFBibTeX XMLCite \textit{A. Švec}, Czech. Math. J. 37(112), 207--230 (1987; Zbl 0634.58037) Full Text: EuDML References: [1] Berger M., Gouduchon P., Mazet E.: Le spectre d’une variété riemannienne. Lecture Notes in Math., vol. 194 (1971), Springer-Verlag. [2] Bochner S., Yano K.: Curvature and Betti numbers. Princeton University Press, 1953. · Zbl 0051.39402 [3] Chavel L: Eigenvalues in Riemannian Geometry. Academic Press, 1984. · Zbl 0551.53001 [4] Iwasaki L, Katase K.: On the spectra of Laplace operator on A*{\(S_n\)). Proc. Japan Acad., 55, Sér. A (1979), 141-145.} · Zbl 0442.58029 · doi:10.3792/pjaa.55.141 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.