# Transversally strictly hyperbolic systems

## TL;DR

This paper introduces a new class of first order systems called transversally strictly hyperbolic systems, analyzing their hyperbolicity properties based on the geometric nature of the characteristic manifold.

## Contribution

The paper defines transversally strictly hyperbolic systems and establishes conditions under which they are strongly hyperbolic, depending on the involutive or symplectic nature of the characteristic manifold.

## Key findings

- Transversally strictly hyperbolic systems are strongly hyperbolic if the characteristic manifold is involutive or symplectic.
- The hyperbolicity properties are more complex when the characteristic manifold is neither involutive nor symplectic.
- An example illustrates the increased complexity in the non-involutive, non-symplectic case.

## Abstract

We consider the Cauchy problem for first order systems. Assuming that the set of the singular points of the characteristic variety is a smooth manifold and the characteristic values are real and semi-simple we introduce a new class which is strictly hyperbolic in the transverse direction to the characteristic manifold. We prove that if the characteristic manifold is either involutive or symplectic then these transversally strictly hyperbolic systems are strongly hyperbolic. On the other hand if the characteristic manifold is neither involutive nor symplectic transversally strictly hyperbolic systems are much more involved which is discussed taking an interesting example.