When a $(1,1)$-tensor generates separation of variables of a certain metric

TL;DR

This paper introduces a method to determine when a (1,1)-tensor field can generate separation of variables for a metric by constructing explicit differential invariants that vanish precisely under this condition.

Contribution

The authors develop an explicit system of differential invariants associated with a (1,1)-tensor that characterizes the existence of a metric enabling variable separation.

Findings

Explicit differential invariants are constructed for (1,1)-tensors.

Invariants vanish if and only if the tensor generates variable separation.

Provides a criterion for metric existence based on tensor invariants.

Abstract

By a $(1,1)$-tensor field $K= K^i_j$, we construct an explicit system of differential invariants that vanish if and only if there (locally) exists a metric for which $K$ generates separation of variables.