Tensor fields of type (0,2) on linear frame bundles and cotangent bundles (Q1589926)

Let \(M\) be a smooth manifold of dimension \(n\geq 2\). The author associates to any \((0,2)\)-tensor field on the frame bundle (Section 2), respectively on the cotangent bundle (Section 4), a global matrix function when a linear connection or a Riemannian metric on \(M\) is given. Next, based on this fact, natural \((0,2)\)-tensor fields on the frame bundle and the cotangent bundle, respectively, are defined and characterized by means of some well known algebraic results (Proposition 3.2 and Theorem 5.1). In the symmetric case, the obtained classification agrees with the one given by Sekizawa and Kowalski-Sekizawa (Remarks 3.2 and 5.1). However, in this paper the author does not make use of the theory of differential invariants.