Nstitutional Critique Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: All

Nstitutional Critique Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: All data are out there and may be provided upon request. Acknowledgments: The authors sincerely thank the reviewers and the editor for their helpful comments and important ideas, which have considerably enhanced the presentation of this paper. The authors are grateful towards the CX-5461 In Vitro Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, for partial financial help. This study is partially supported by the Research Administration Centre, Chiang Mai University. Conflicts of Interest: The authors declare no conflict of interest.mathematicsArticleOn Invariant operations on a Manifold using a Linear Connection and an OrientationAdri Gordillo-Merino , Ra Mart ez-Boh quez and JosNavarro-Garmendia Departamento de Matem icas, Universidad de Extremadura, Avenida de Elvas s/n, 06006 Badajoz, Spain; [email protected] (A.G.-M.); [email protected] (R.M.-B.) Correspondence: [email protected]: We prove a theorem that describes all feasible tensor-valued all-natural operations in the presence of a linear connection and an orientation in terms of specific linear representations of your unique linear group. As an application of this outcome, we prove a characterization with the torsion and curvature operators as the only organic operators that satisfy the Bianchi identities. Key phrases: organic tensors; linear connections; torsion tensor; curvature operator PACS: 53A55; 58A1. Introduction Since the really early days of differential geometry, the idea of natural operation played a mayor part in the development the theory. As an instance, let us point out the applications of this notion of naturalness within the inception of common relativity (cf. [1]). Inside the course on the years, there also appeared some striking mathematical benefits, like Gilkey’s characterization of Pontryagin forms on Riemannian manifolds [2,3] or his proof on the uniqueness on the Chern auss onnet formula [4]. By the finish from the final century, the contemporary improvement of this Tridecanedioic acid web theory was summarized within the monograph by Kol-Michorr Slov [5]. That book contained all the most important results and procedures that had been recognized so far, and therefore became the regular reference within the subject since then. On the other hand, the notion of covariance or naturalness is, in some sense, ubiquitous in physics and mathematics. For that cause, the renewed interest in this theory of organic operations that has been raised in recent years just isn’t surprising, using the look of new outcomes and applications in speak to geometry [6], homotopy theory [7,8], Riemannian and K ler geometry [92], common relativity [13], or quantum field theory [14,15]. In this paper, we focus our focus on the vector space of tensor-valued natural operations that could be performed inside the presence of a linear connection and an orientation. Our principal result, Theorem 8, establishes that such a vector space is isomorphic for the space of invariant maps among specific linear representations of the unique linear group. Thus, the description of these spaces can, in certain cases, be fully achieved making use of classical invariant theory. As an example of this philosophy, within the final section, we characterize the torsion as well as the curvature as the only all-natural tensors satisfying the Bianchi identities (Corollary 13 and Theorem 15). These benefits generalize analogous statements that were not too long ago proven i.