A theory of generally invariant lagrangians for the metric fields. I

Abstract

The second-order generally invariant Lagrangians for the metric fields are studied within the framework of the Ehresmann theory of jets. Such a Lagrangian is a function on an appropriate fiber bundle whose structure group is the groupL _n ³ of invertible 3-jets with source and target at the origin 0 of the real,n-dimensional Euclidean spaceR ⁿ, and whose type fiber is the manifold T_n ²(R^n* ⊙R ^n*) of 2-jets with source at 0 εR ⁿ and target in the symmetric tensor productR ^n* ⊙ R^n*. Explicit formulas for the action ofL _n ³ onT _n ²(R^n* ⊙R ^n*) are considered, and a complete system of differential identities for the generally invariant Lagrangians is obtained.