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We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differentialoperators} with coefficients in general Hermitian vector bundles on Riemannianmanifolds, stressing a coordinate free approach that uses connections (whichare typically denoted $\nabla$). These concepts arise naturally from PartialDifferential Equations, including some that are formulated on plain Euclideandomains, such as the weighted Sobolev spaces used to study PDEs on singulardomains. We prove several basic properties of the $\nabla$-Sobolev spaces andof the $\nabla$-differential operators on general manifolds. For instance, weprove mapping properties for our differential operators and independence of the$\nabla$-Sobolev spaces on the choices of the connection $\nabla$ with respectto totally bounded perturbations. We introduce a {\em Fr\'echet finitenesscondition} (FFC) for totally bounded vector fields, which is satisfied, forinstance, by open subsets of manifolds with bounded geometry. When (FFC) issatisfied, we provide several equivalent definitions of our $\nabla$-Sobolevspaces and of our $\nabla$-differential operators. We examine in more detailthe particular case of domains in the Euclidean space, including the case ofweighted Sobolev spaces. We also introduce and study the notion of a {\em$\nabla$-bidifferential} operator (a bilinear version of differentialoperators), obtaining results similar to those obtained for$\nabla$-differential operators. Bilinear differential operators are necessaryfor a global, geometric discussion of variational problems. We tried to writethe paper so that it is accessible to a large audience.

Sobolev spaces and $\nabla$-differential operators on manifolds I: basic properties and weighted spaces