Natural variational principles on Riemannian manifolds. (English) Zbl 0565.58019

Anderson, Ian M.

Ann. Math. (2) 120, 329-370 (1984).

The inverse problem in the calculus of variations is to determine when a given system of differential equations are the Euler-Lagrange equations for a variational problem defined by a Lagrangian L. If the Lagrangian L is invariant under the action of a Lie group G, then G is also a symmetry group of the Euler-Lagrange equations G. Thus the equivariant inverse problem arises, viz. if T is an invariant differential operator, then when is T the Euler-Lagrange operator of an invariant Lagrangian?
In this paper the equivariant inverse problem is solved for natural differential operators on Riemannian structures, i.e. operators \(T=T[j^ kg]\) which are defined upon the k-jet of a Riemannian metric and which satisfy the invariance condition \(T[j^ k[f^*(g)]=f^*T[j^ k(g)]\) for all local diffeomorphisms f. It is shown that the obstructions to the construction of natural Lagrangians, \(L=L[j^{\ell}(g)]\) for such natural operators T are generated by the secondary characteristic classes of Chern-Simon.
In particular, on even dimensional manifolds there are no obstructions while on 3 manifolds the so-called Cotton tensor \(C^{ij}=\epsilon^{ihk}R^ j_{h| k}+\epsilon^{jhk}R^ i_{h| k},\) where \(R^ j_ h\) is the Ricci tensor, is the only natural tensor which is derivable from a variational principle but not an invariant variational principle.

MSC:

58E30

Variational principles in infinite-dimensional spaces

Keywords:

characteristic and secondary characteristic classes; inverse problem in the calculus of variations; Euler-Lagrange equations; natural tensor