Operator-valued tensors on manifolds

In this paper we try to extend geometric concepts in the context of operator valued tensorsý. ýTo this endý, ýwe aim to replace the field of scalars R by self-adjoint elements of a commutative C⋆-algebraý, ýand reach an appropriate generalization of geometrical concepts on manifoldsý. ýFirstý, ýwe put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metricsý. ýThený, ýin this new geometryý, ýsome essential concepts of Riemannian geometry such as curvature tensorý, ýLevi-Civita connectioný, ýHodge star operatorý, ýexterior derivativeý, ýdivergence,..ý. ýwill be considered.