Natural transformations of symmetric affine connections on manifolds to metrics on linear frame bundles: a classification

A new class of metrics on the cotangent bundle

“…In the particular case the metric g f is tensor field of type (0, 2) on T * M . It follows that g f is completely determined by its formulas (10), (11) and (12). By means of (1) and (2), the complete lift X C of X ∈ 1 0 (M ) is given by…”

“…In the field, one of the first works which deal with the cotangent bundles of a manifold as a Riemannian manifold is that of Patterson, E.M. and Walker, A.G. [7], who constructed from an affine symmetric connection on a manifold a Riemannian metric on the cotangent bundle, which they call the Riemann extension of the connection. A generalization of this metric had been given by Sekizawa, M. [12] in his classification of natural transformations of affine connections on manifolds to metrics on their cotangent bundles, obtaining the class of natural Riemann extensions which is a 2-parameter family of metrics, and which had been intensively studied by many authors. On the other hand, inspired by the concept of g-natural metrics on tangent bundles of Riemannian manifolds, F. Agca considered another class of metrics on cotangent bundles of Riemannian manifolds, that he callad g-natural metrics [1].…”

Berger Type Deformed Sasaki Metric and Harmonicity on the Cotangent Bundle

“…In the field, one of the first works which deal with the cotangent bundles of a manifold as a Riemannian manifold is that of Patterson, E. M., Walker, A. G. [9] who constructed from an affine symmetric connection on a manifold a Riemannian metric on the cotangent bundle, which they call the Riemann extension of the connection. A generalization of this metric had been given by Sekizawa, M. [12] in his classification of natural transformations of affine connections on manifolds to metrics on their cotangent bundles, obtaining the class of natural Riemann extensions which is a 2-parameter family of metrics, and which had been intensively studied by many authors. On the other hand, inspired by the concept of g-natural metrics on tangent bundles of Riemannian manifolds, Agca, F. considered another class of metrics on cotangent bundles of Riemannian manifolds, that he callad g-natural metrics [1].…”

Lifting to the r-frame bundle by means of connections

“…[1], [6]). In [10], using rather complicated computations in local coordinates, M. Sekizawa obtained an interesting classification of all first order Mf m -natural operators A : Q T (0,2) P 1 transforming classical linear connections ∇ on m-manifolds M into tensor fields A(∇) of type (0, 2) on the linear frame bundle LM = P 1 M of M . A well-known example of an Mf m -natural op-erator A : Q QP r is the so-called complete lift in the sense of A. Morimoto [8] (see also [2]) of classical linear connections to the rth order frame bundle P r M (which is an open subbundle in the bundle T r m M of (m, r)velocities).…”

“…A well-known example of an Mf m -natural op-erator A : Q QP r is the so-called complete lift in the sense of A. Morimoto [8] (see also [2]) of classical linear connections to the rth order frame bundle P r M (which is an open subbundle in the bundle T r m M of (m, r)velocities). In [7], using a normal coordinate technique, the second author extended (in a very simple way) the classification of [10] to all Mf m -natural operators A : Q T (p,q) P r , and in particular obtained a classification of all Mf m -natural operators A : Q QP r . In [4], adapting the technique from [7], we classified all Mf m -natural operators A : T (p,q) × Q T (p 1 ,q 1 ) P 1 transforming tensor fields τ of type (p, q) on M and classical linear connections ∇ on M into tensor fields A(τ, ∇) of type (p 1 , q 1 ) on LM .…”