1 CURVATURE AND CAYLEY-MENGER DETERMINANTS Author: Dr Franck Delplace ESI Group Scientific Committee, 100 Avenue de Suffren, Paris, France fr.delplace@gmail.com ABSTRACT: In this paper, we demonstrated that the hydraulic diameter used in fluid mechanics can be related to Cayley-Menger determinants ratios. The relationship first established for 2D shapes is extended to n-shapes. Considering hydraulic diameter as an intrinsic characteristic of geometries, a definition of their global curvature is proposed. Consequences in fluid mechanics, particularly for flows stability are discussed giving interesting perspectives for Navier-Stokes equation resolution. 1. INTRODUCTION As showed by H. Poincaré [1], solids curvature is of major importance when studying diffusion phenomena. Riemannian curvature, as defined by Ricci tensor (used in General Relativity theory) and studies on Ricci flow in Riemannian manifolds with given metric tensor, gives a diffusion equation analogous to heat diffusion [2]. Diffusion phenomena concern both scalar fields (temperature for Fourier equation or solute concentration for Fick equation) and vectors fields (velocity and momentum in Navier-Stokes equation). Recently, Delplace and Srivastava [3] insisted on the importance of curvature in the case of velocity fields encountered in fluid mechanics for the flow of viscous liquids. Based on experimental results obtained in laminar flow conditions [4-7], Delplace [8] proposed to define the famous Reynolds dimensionless number, usually considered as the ratio of inertia to viscous forces, in a new way: the ratio of pipe or container curvature and fluid streamlines curvature. Moreover, from a balance between pressure drop and viscous friction dissipation in a pipe of arbitrary cross-section shape, this author [9] proposed a new definition of curvature available for regular polygons, polyhedra and n-gones. This approach can be considered very similar to the first demonstration of isoperimetric inequality using soap bubbles equilibrium under superficial tensions. Furthermore, this definition of curvature involves hydraulic radius well known in fluid mechanics and used to define Reynolds number as the ratio of two curvatures as reported above. It is important to insist that all this work is based on a great number of experimental, numerical and analytical results giving a strong basis to Delplace’s approach. Consequently, the use of hydraulic radius as a mean to calculate curvature of surfaces, regular polyhedra and even n-gones appears to be a very interesting method for important challenges in mathematics about Riemannian manifolds like Ricci flow. But, even if hydraulic radius is well defined in fluid mechanics and well in agreement with isoperimetric inequality [9], there is always a lake of mathematical link with intrinsic characteristics of complex geometries i.e. their metric and associated quantities.