Author/Authors :
Feizabadi H. نويسنده Faculty of Mathematics & Computer Science, Amirkabir University of Technology, Tehran, Iran. , Boroojerdian N. نويسنده Faculty of Mathematics & Computer Science, Amirkabir University of Technology, Tehran, Iran.
Abstract :
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.
