Defining map

If is an embedded submanifold, a smooth map is called a defining map if is a regular level set, i.e. if for a regular point .

Local defining map

A map is called locally defining if for some subset and is a regular level set.

Motivation

We call this a defining map, because based on the regular value theorem this map along with the regular value would define a submanifold.

Relation to tangent space of submanifold

For a submanifold , if is a local defining map for , then for every .

Proof

todo