# Remarks on natural differential operators with tensor fields

## TL;DR

This paper investigates natural differential operators that map pairs of tensor fields to a tensor field, establishing that all bilinear operators are first-order and providing a complete classification in specific cases.

## Contribution

It proves that all bilinear natural differential operators are of order one and offers a comprehensive classification in various concrete scenarios.

## Key findings

- All bilinear operators are of order one.
- Complete classification of such operators in specific cases.
- Provides foundational understanding of natural differential operators with tensor fields.

## Abstract

We study natural differential operators transforming two tensor fields into a tensor field. First, it is proved that all bilinear operators are of order one, and then we give the full classification of such operators in several concrete situations.