nLab
positive n-form

Idea

Let $X$ be an oriented $n$-dimensional differentiable manifold. The $n$-forms on $X$ have a partial order $\le$, which we define (via subtraction) through the notion of positive $n$-form. This makes the line bundle of $n$-forms on $X$ into an oriented line bundle?.

Definitions

An $n$-form $\omega$ is positive semidefinite or just positive (denoted $\omega \ge 0$) if the following equivalent conditions hold:

• the integral of $\omega$ on any open submanifold? of $X$ is nonnegative;
• the coordinate of $\omega$ in any oriented local coordinate system on $X$ is nonnegative;
• the value of $\omega$ at any point $p$, when applied to any positively oriented collection of $n$ tangent vectors at $p$, is nonnegative.

Supposing $X$ is unoriented, an $n$-pseudoform $\omega$ is positive semidefinite if the following equivalent conditions hold:

• the integral of $\omega$ on any open submanifold? of $X$ is nonnegative;
• the coordinate of $\omega$ in any local coordinate system on $X$ is nonnegative;
• the value of $\omega$ at any point $p$, when applied relative to either local orientation $o$ at $p$ to any $o$-oriented collection of $n$ tangent vectors at $p$, is nonnegative.

Arguably, positivity of pseudoforms is fundamental; an orientation allows one to interpret forms as pseudoforms.

A (pseudo)-form is positive definite (denoted $\omega >0$) 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 $X$.

Properties

If $X$ is oriented, then every $n$-form $\omega$ has an absolute value $\mid \omega \mid$, which is a positive $n$-form. However, even if $\omega$ is smooth, still $\mid \omega \mid$ may only be continuous. However, if $\omega$ is nondegenerate, then not only will $\mid \omega \mid$ be also nondegenerate, it will be smooth (if $\omega$ is). Even if $X$ is unoriented, still any $n$-form or $n$-pseudoform $\omega$ will have a positive $n$-pseudoform $\mid \omega \mid$ as absolute value.