nLab
positive n-form

Positive nn-forms

Idea

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

Definitions

An nn-form ω\omega is positive semidefinite or just positive (denoted ω0\omega \geq 0) if the following equivalent conditions hold:

  • the integral of ω\omega on any open submanifold? of XX is nonnegative;
  • the coordinate of ω\omega in any oriented local coordinate system on XX is nonnegative;
  • the value of ω\omega at any point pp, when applied to any positively oriented collection of nn tangent vectors at pp, is nonnegative.

Supposing XX is unoriented, an nn-pseudoform ω\omega is positive semidefinite if the following equivalent conditions hold:

  • the integral of ω\omega on any open submanifold? of XX is nonnegative;
  • the coordinate of ω\omega in any local coordinate system on XX is nonnegative;
  • the value of ω\omega at any point pp, when applied relative to either local orientation oo at pp to any oo-oriented collection of nn tangent vectors at pp, is nonnegative.

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

A (pseudo)-form is positive definite (denoted ω>0\omega \gt 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 XX.

Properties

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

Revised on January 24, 2013 09:33:18 by Toby Bartels (98.19.39.195)