Higher Order Hessian Structures on Manifolds

[摘要] In this paper we define 𝑛th order Hessian structures on manifolds and study them. In particular, when 𝑛=3, we make a detailed study and establish a one-to-one correspondence between third-order Hessian structures and a certain class of connections on the second-order tangent bundle of a manifold. Further, we show that a connection on the tangent bundle of a manifold induces a connection on the second-order tangent bundle. Also we define second-order geodesics of special second-order connection which gives a geometric characterization of symmetric third-order Hessian structures.

[学科分类] 数学（综合）

[关键词] Hessian structure;connection;geodesic.