Operator-valued tensors on manifolds

In this paper we try to extend geometric concepts in the context of operator valued tensors. To this end, we aim to replace the eld of scalars R by self-adjoint elements of a commutative C*-algebra, and reach an appropriate generalization of geometrical concepts on manifolds. First, we put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metrics. Then, in this new geometry, some essential concepts of Riemannian geometry such as curvature tensor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,... will be considered.