\(L^ 2\) vector bundle valued forms and the Laplace-Beltrami operator (Q1087206)

Let (M,g) be a smooth orientable connected Riemannian manifold of dimension n. We consider a smooth vector bundle \(\xi\) : \(V\to M\) of finite rank m. We assume that on the vector bundle \(\xi\) there are a positive definite inner product \(< >\) and a connection \(\nabla\) which are related as follows \(X<w,\phi >=<\nabla_ Xw,\phi >+<w,\nabla_ X\phi >\), for all vector fields X and all sections w,\(\phi\) of \(\xi\). The triple \(\xi\), \(<, >\), \(\nabla\) is called a Riemannian vector bundle. We shall denote by C(\(\xi)\) or C(V) the vector space of smooth sections of \(\xi\). The smooth sections of \(\Lambda^ pT^*(M)\otimes V\to M\) are called p-forms on M with values in \(\xi\), and we shall denote the space of \(\xi\)-valued p-forms by \(E^ p(\xi)\) or \(E^ p(C)\). The aim of the present paper is to study some spectral problems of the Laplace operator and to examine the Hodge orthogonal decomposition for vector valued differential forms.