%0 Journal Article %T Operator-valued tensors on manifolds %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A ‎Feizabadi, H. %A Boroojerdian, N. %D 2016 %\ 10/01/2016 %V 42 %N 5 %P 1259-1277 %! Operator-valued tensors on manifolds %K Operator-valued tensors‎ %K operator-valued semi-Riemannian metrics‎ %K ‎Levi-Civita connection‎ %K curvature‎ %K ‎Hodge star operator %R %X ‎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. %U http://bims.iranjournals.ir/article_879_44f3202eefb900362bb1960d135193a5.pdf