Rank of Jacobi operator and existence of quadratic parallel differential form, with applications to geometry of almost para-contact metric manifolds

TL;DR

This paper investigates the relationship between the rank of the Jacobi operator and the existence of quadratic parallel differential forms in pseudo-Riemannian manifolds, with applications to almost para-contact metric manifolds, providing algebraic criteria for such structures.

Contribution

It introduces an algebraic method to determine the existence of non-isotropic vector fields with maximal rank Jacobi operator using a canonical differential form.

Findings

Existence of such vector fields obstructs non-trivial second-order symmetric parallel tensors.

Presence of these vector fields implies the manifold is locally non-reducible.

An effective algebraic algorithm is provided for identifying these vector fields.

Abstract

It is established that the existence of non-isotropic vector field which Jacobi operator of maximal rank is an obstacle for the existence of non-trivial second-order symmetric parallel tensor field. In turns out that presence of such obstacle follows that manifold as pseudo-Riemannian manifold is locally non-reducible. In particular result can be applied directly to known classes of almost (para-) contact metric manifolds when considered Jacobi operator of characteristic vector field has maximal rank. There is effective algorithmic procedure which resolves the problem of existence of such vector field in pure algebraic way - there is canonically defined homogeneous differential form, with coefficients determined purely by the coeffcients of curvature operator - such that non-isotropic vector field has non-degenerate Jacobi operator if and only if it is non-zero of this form.