  Recently, Hessian geometry - an extension of information geometry - has
emerged as a framework to naturally connect the geometries appearing in the
theory of chemical reaction networks (CRN) to their inherent thermodynamical
and kinetic properties. This framework is used in this letter to derive
multivariate thermodynamic uncertainty relations (TUR) for CRN. The matrices
featured in the TUR are shown to be representations of Riemmanian metric
tensors, whereby one tensor characterizes the pseudo entropy production rate
and the other the current fluctuations. It is shown that the latter tensor is a
restriction of the former one to a linear subspace in the flux tangent space.
Therefore, in addition to clarifying the geometric origin of TUR in CRN, the
Hessian geometric setup yields a characterization of the error term in the TUR
as the norm of a linear subspace component of the flux vector and thus
characterizes the fluxes where TUR become equalities.
