Abstract
If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T*M given by v → g(v, –) between the tangent TM and the cotangent T*M bundles of M. In the present note first we generalize this isomorphism to the one J<sup>r</sup>TM → J<sup>r</sup>T*M between the r-th order prolongation J<sup>r</sup>TM of tangent TM and the r-th order prolongation J<sup>r</sup>T*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps D<sub>M</sub>(g) : J<sup>r</sup>TM → J<sup>r</sup>T*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.
Highlights
All manifolds are smooth, Hausdorff, finite dimensional and without boundaries
If g is a Riemannian metric on a manifold M, there is the base preserving vector bundle isomorphism ig : T M → T ∗M given by ig(v) = g(v, −), v ∈ TxM, x ∈ M
In the fourth section we consider the problem of describing all Mfmnatural operators D : Riem Hom(JrT, JrT ∗) transforming Riemannian metrics g on m-dimensional manifolds M into base preserving vector bundle maps DM (g) : JrT M → JrT ∗M → (JrT ∗M)
Summary
Hausdorff, finite dimensional and without boundaries. Maps are assumed to be smooth, i.e. of class C∞. In the fourth section we consider the problem of describing all Mfmnatural operators D : Riem Hom(JrT, JrT ∗) transforming Riemannian metrics g on m-dimensional manifolds M into base preserving vector bundle maps DM (g) : JrT M → JrT ∗M. The r-th order prolongation of tangent bundle is a functor JrT : Mfm → VB sending any m-manifold M into JrT M and any embedding φ : M1 → M2 of two manifolds into JrT φ : JrT M1 → JrT M2 given by J rT φ(jxrX) = jφr (x)φ∗X, where X ∈ X (M1) and φ∗X = T φ ◦ X ◦ φ−1 is the image of a vector field X by φ. An Mfm-natural operator D : Riem Hom(JrT, JrT ∗) transforming Riemannian metrics g on m-dimensional manifolds M into base preserving vector bundle maps DM (g) : JrT M → JrT ∗M is a system D = {DM }M∈obj(Mfm) of regular operators.
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