# Max-decreasing trajectory

From Diffgeom

*This article defines a property that can be evaluated for a trajectory on the space of functions on a manifold*

## Definition

Let be a manifold and be a function , where:

- denotes the time parameter, and varies in
- denotes the spatial parameter, and varies in

In other words, is a trajectory (or path) in the space of all functions from to .

Then, is said to be **max-decreasing** if the function:

is a monotone decreasing function. (The function defined above is termed the timewise-max function for ).

The corresponding notion is of a **min-increasing trajectory** -- viz a trajectory where the minimum (or infimum) keeps increasing.