1
$\begingroup$

I recently learnt about s-densities:

http://en.wikipedia.org/wiki/Density_on_a_manifold#s-densities_on_a_vector_space

For simplicity suppose that the vector space in this definition is \R. The prime example of a 1-density is certainly the one form dx. Each 1-density integrates to a translation-invariant measure on \R.

Is there a similar relation between s-densities and conformal measures?

Improve this question
$\endgroup$
3
  • $\begingroup$ What's a conformal density? $\endgroup$
    – Deane Yang
    Commented Jan 24, 2012 at 8:45
  • $\begingroup$ @Deane Yang: I think it should read conformal measure. In our situation an s-conformal measure on \R is a measure m on \R such that m(ax) = |a|^s m(x) for all non-zero scalars a and all x on \R (or subsets of the Borel sigma-algebra on \R). $\endgroup$
    – Otmar Möller
    Commented Jan 24, 2012 at 18:14
  • $\begingroup$ What is an example of an $s$-conformal measure that is not of the form $c|x|^{s-1}\,dx$? $\endgroup$
    – Deane Yang
    Commented Jan 24, 2012 at 18:20

1 Answer 1

Reset to default
2
$\begingroup$

I am not sure what exactly a “conformal measure” is, but densities do play a prominent role in the definition of conformal structures.

Consider a smooth manifold M. A conformal structure on M is a Riemannian metric on the weightless cotangent bundle T*0M=T*M⊗Dens−1/dim M(M). Expanding this definition yields the usual notion of an equivalence class of ordinary Riemannian metrics.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .