Let be a smooth manifold of dimension .
Definition. If is an embedded submanifold, a smooth map such that is a regular level set of is called a defining map for . In other words,
for some point . In particular, if (so that is a real-valued or vector-valued function), it is usually called a defining function.
Example 1. The sphere is an embeded submanifold of . The sphere is easily seen to be a regular level set of the function given by since vanishes only at the origin.
Definition. More generally, if is an open subset of and is a smooth map such that is a regular level set of , then is called a local defining map (or local defining function) for .
Example 2. The smooth map given by
is an immersion of into whose image, denoted by , is the doughnut-shaped shape surface obtained by revolving the circle around the -axis, a point is in if and only if it satisfies where is the distance from the -axis. Thus is the zero set of the function which is smooth on minus the -axis. A straightforward computation shows that $lated d\Psi$ does not vanish on , so is a global defining function for .