Timewise-average function

This article defines a scalar function on a manifold, viz a function from the manifold to real numbers. The scalar function may be intrinsic or defined in terms of some other structure/functions

Definition

Let $M$ be a differential manifold equipped with a measure over which we can integrate.

Let $u:\mathbb{R}\times M\to \mathbb{R}$ be any continuous function. Then the timewise-average function of $u$ is a map $f:\mathbb{R}\to \mathbb{R}$ defined by:

$f(t)=\frac{{\int }_{M}u(t,x)dx}{{\int }_{M}dx}$

For instance, the above notion makes sense if we embed the manifold in Euclidean space and look at the induced measure, or in a Riemannian manifold where we take the naturally induced measure.

Related notions

• Timewise-max function
• Timewise-min function