  Extensions (entropies) play a central role in the theory of hyperbolic
conservation laws by providing intrinsic selection criteria for weak solutions.
For a given hyperbolic system u_t+f(u)_x=0, a standard approach is to analyze
directly the second order PDE system for the extensions. Instead we find it
advantageous to consider the equations satisfied by the lengths beta^i of the
right eigenvectors r_i the Jacobian matrix Df, as measured with respect to the
inner product defined by an extension. Our geometric formulation provides a
natural and systematic approach to existence of extensions. By prescribing the
eigen-fields r_i our results automatically apply to all systems with the same
eigen-frame. The equations for the lengths beta^i form a first order
algebraic-differential system (the beta-system) to which standard integrability
theorems can be applied. The size of the set of extensions follows by
determining the number of free constants and functions present in the general
solution to the beta-system. We provide a complete breakdown of the various
possibilities for systems of three equations, as well as for rich hyperbolic
systems of any size.
