Higher order Hessian structures on manifolds (Q2572423)

Assume that \(M\) is a differentiable manifold and denote by \(T^{n} M\) its tangent bundle of order \(n\). In the paper under review the author defines Hessian structures of order \(n\) on \(M\). Special attention is paid to the case \(n=3\). It is shown that there exists a one-to-one correspondence between third-order Hessian structures and a certain class of linear connections on the second-order tangent bundle of \(M\). Further, it is shown that a linear connection on \(TM\) induces a linear connection on \(T^{2}M\). Using a special second-order connection the author defines a class of second order geodesics. It turns out that this class leads to a characterization of symmetric third-order Hessian structures.