Let be an oriented -dimensional differentiable manifold. The -forms on have a partial order , which we define (via subtraction) through the notion of positive -form. This makes the line bundle of -forms on into an oriented line bundle?.
An -form is positive semidefinite or just positive (denoted ) if the following equivalent conditions hold:
Supposing is unoriented, an -pseudoform is positive semidefinite if the following equivalent conditions hold:
Arguably, positivity of pseudoforms is fundamental; an orientation allows one to interpret forms as pseudoforms.
A (pseudo)-form is positive definite (denoted ) if it is also nondegenerate (nowhere zero), or equivalently if the conditions above hold with “strictly positive” in place of “nonzero” (taking integrals over only inhabited submanifolds). A positive definite form can be interpreted as a volume (pseudo)-form on .
If is oriented, then every -form has an absolute value , which is a positive -form. However, even if is smooth, still may only be continuous. However, if is nondegenerate, then not only will be also nondegenerate, it will be smooth (if is). Even if is unoriented, still any -form or -pseudoform will have a positive -pseudoform as absolute value.