The Legendre maps from two Lagrangians or from a Lagrangian and a p-form

Abstract

Let \(\mathcal{FM}_{m,n}\) denote the category of fibered manifolds with \(m\)-dimensional bases and \(n\)-dimensional fibres and their fibered local diffeomorphisms. We prove that  if \(m,n\) and \(s\) are positive integers, then any \(\mathcal{FM}_{m,n}\)-natural operator \(C\) transforming tuples \((\lambda_1,\lambda_2)\) of Lagrangians \(\lambda_1,\lambda_2:J^sY\to\bigwedge ^mT^*M\) on \(\mathcal{FM}_{m,n}\)-objects \(Y\to M\) into Legendre maps \(C(\lambda_1,\lambda_2):J^{s}Y\to S^sTM\otimes V^*Y\otimes\bigwedge^m T^*M\) on \(Y\) is of the form \(C(\lambda_1,\lambda_2)=c_1\Lambda(\lambda_1)+c_2\Lambda(\lambda_2)\), \(c_1,c_2\in\mathbf{R}\), where \(\Lambda\) is the Legendre operator. We also prove that if \(m,n,s\) and \(p\) are  positive integers, then any \(\mathcal{FM}_{m,n}\)-natural operator \(C\) transforming tuples \((\lambda,\eta)\) of Lagrangians \(\lambda:J^sY\to\bigwedge ^mT^*M\) and \(p\)-forms \(\eta\in \Omega^p(M)\) into Legendre maps \(C(\lambda,\eta):J^{s}Y\to S^sTM\otimes V^*Y\otimes\bigwedge^m T^*M\) is of the form \(C(\lambda,\eta)=c\Lambda(\lambda)\), \(c\in\mathbf{R}\), where \(\Lambda\) is the Legendre operator.