A new approach for studying wave propagation phenomena in an inviscid gas i
s presented. This approach can be viewed as the extension of the method of
characteristics to the general case of unsteady multidimensional flow. A fa
mily of spacetime manifolds is found on which an equivalent one-dimensional
(1-D) problem holds. Their geometry depends on the spatial gradients of th
e flow, and they provide, locally, a convenient system of coordinate surfac
es for spacetime. In the case of zero-entropy gradients, functions analogou
s to the Riemann invariants of 1-D gas dynamics can be introduced. These ge
neralized Riemann invariants are constant on these manifolds and, thus, the
manifolds are dubbed Riemann invariant manifolds (RIM). Explicit expressio
ns for the local differential geometry of these manifolds can be found dire
ctly from the equations of motion. They can be space-like or time-like, dep
ending on the flow gradients. This theory is used to develop a second-order
unsplit monotonic upstream-centered scheme for conservation laws (MUSCL)-t
ype scheme for the compressible Euler equations. The appropriate RIM are tr
aced back in time, locally, in each cell. This procedure provides the state
s that are connected with equivalent 1-D problems. Furthermore, by assuming
a linear variation of all quantities in each computational cell, it is pos
sible to derive explicit formulas for the states used in the 1-D characteri
stic problem.