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 field of scalars $ mathbb{R} $ by self-adjoint elements of a commutative $ C^star $-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.