# Operator-valued tensors on manifolds

Document Type: Research Paper

Authors

1 Faculty of Mathematics & Computer Science‎, ‎Amirkabir University of Technology‎, ‎Tehran‎, ‎Iran.

2 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.

Keywords

Main Subjects

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### History

• Receive Date: 23 November 2014
• Revise Date: 15 August 2015
• Accept Date: 15 August 2015