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Vector fields on hyperspaces. (English) Zbl 0634.58037

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

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
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References:

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