Bernhard Riemann 1861 revisited: existence of flat coordinates for an arbitrary bilinear form

TL;DR

This paper extends classical results by providing explicit criteria for when a general bilinear form can be expressed with constant entries in some local coordinates, broadening the understanding of flatness beyond symmetric cases.

Contribution

It offers necessary and sufficient conditions for a general (not necessarily symmetric or non-degenerate) bilinear form to be flat, including degenerate cases.

Findings

Derived explicit conditions for flatness of bilinear forms.

Extended flatness criteria to degenerate and non-symmetric forms.

Generalized classical results of Riemann and Darboux.

Abstract

We generalize the celebrated results of Bernhard Riemann and Gaston Darboux: we give necessary and sufficient conditions for a bilinear form to be flat. More precisely, we give explicit necessary and sufficient conditions for a tensor field of type (0,2) which is not necessary symmetric or skew-symmetric, and is possibly degenerate, to have constant entries in a local coordinate system.