Does integration require a metric?

[friend]

I think I remember reading somewhere that all the machinery of manifolds and a metric needed to be established first before the integral and the differential of calculus had any meaning. Am I remembering wrong? Is there such a thing as coordinate independent integration or differentiation? Thanks.

 

[Number Nine]

Are you familiar with the concept of a differential form?

 

[micromass]

I think I remember reading somewhere that all the machinery of manifolds and a metric needed to be established first before the integral and the differential of calculus had any meaning. Am I remembering wrong? Is there such a thing as coordinate independent integration or differentiation? Thanks.

No, a metric is not needed to define integration on manifolds. Like Number Nine suggested, you should research differential forms. These are objects which can be integrated. And the integration can be defined coordinate independent.

A good reference is "Introduction to smooth manifolds" by Lee.

 

[lavinia]

As micromass and number 9 have said integration is defined without a metric in terms of differential forms.

However, integrals are often expressed in terms of areas,lengths, and volumes. These ideas require a metric.

 

[mathman]

The minimum requirement needed to define an integral over a set is that there be a measure defined on it. No further structure is necessary.
.

 

[friend]

The minimum requirement needed to define an integral over a set is that there be a measure defined on it. No further structure is necessary.
.

Does that mean that differential forms are a kind of measure?

And since forms are dual to vectors in the tangent space, does that mean that one should be able to integrate vectors as well as forms?

 

[mathman]

Does that mean that differential forms are a kind of measure?

And since forms are dual to vectors in the tangent space, does that mean that one should be able to integrate vectors as well as forms?

When you are talking about integration over real n-dimensional spaces, the measure is usually assumed to be that developed from area or volume, etc. Integration involves functions (diff. forms, vectors, or anything else) defined on the spaces.

 

[micromass]

Does that mean that differential forms are a kind of measure?

A differential form induces a measure in a standard way. See Lang's "Real and functional analysis", chapter XXIII, section 3.

 

[friend]

A differential form induces a measure in a standard way. See Lang's "Real and functional analysis", chapter XXIII, section 3.

And is it true that EVERY measure is a "distribution". I see written on wikipedia.org that, "measures can be taken to be a special kind of distribution". So are all measures interchangeable with a distributions, and visa versa? Can we integrate a distribution like a measure, like a form?

 

[friend]

And is it true that EVERY measure is a "distribution". I see written on wikipedia.org that, "measures can be taken to be a special kind of distribution". So are all measures interchangeable with a distributions, and visa versa? Can we integrate a distribution like a measure, like a form?

This is an interesting question for me. I wonder if a distribution used in quantum theory might be taken as a measure or metric in the sense of general relativity. Can the rules that manipulate distributions in QM somehow be re-interpreted to give rules on how to manipulate the metric for GR?