On jets of surfaces (Q1210114)

In the author's terminology, the \(k\)-jet bundle of surfaces over a manifold \(M\) is the jet space \(J^ k_ 0(\mathbb{R}^ 2,M)\), which is usually denoted by \(T^ k_ 2M\) and called the bundle of 2-dimensional \(k\)-velocities on \(M\). First the author constructs a canonical involutive automorphism \(\alpha\) of \(J^ 1_ 0(\mathbb{R}^ 2,J^ 1_ 0(\mathbb{R}^ 2,M))\). (This map is a special case of a more general isomorphism \(T^ r_ kT^ s_ \ell M\to T^ s_ \ell T^ r_ kM\) constructed by the reviewer [Math. Nachr. 69, 297-306 (1975; Zbl 0318.53034)]). Then the author deduces that the canonical injection of \(J^ 2_ 0(\mathbb{R}^ 2,M)\) into \(J^ 1_ 0(\mathbb{R}^ 2,J^ 1_ 0(\mathbb{R}^ 2,M))\) can be characterized as the invariant set of \(\alpha\).