Defining the integral of 1-forms without parametrization

[mma]

We saw in the thread https://www.physicsforums.com/showthread.php?t=238464" that arc length that is usually defined by taking an arbitrary parametrisation of the curve as

$$l(\gamma)=\int_{0}^{1} {|\dot\gamma(t)|} dt$$​

can be defined also by avoiding parametrization, introducing the notion of the distance of points as

$$d(x,y) = \sup\{|a(y)-a(x)| : a \in C(M), \Vert{\mathrm{grad} a\Vert _\infty \leq 1\}$$​

where

$$\Vert{\mathrm{grad} a\Vert _\infty = sup\{\mathrm{grad} a|_x: x \in M\}$$​

(see equation 3.5 on page 34 of http://ncg.mimuw.edu.pl/index.php?option=com_content&task=view&id=148&Itemid=98", thanks gel for finding it).

The defnition of the integral of an 1-form over a curve is also defined usually by taking a parametrization of the curve:

$$\int_\gamma \omega =\int_{0}^{1} \omega(\dot\gamma(t)) dt$$​

I wondered if we can find a definition of this integral also by avoiding the parametrization.

 

[yyat]

I only see an alternative definition of the distance on a Riemannian manifold, not of the arc-length of a curve, am I missing something?
Also, the only definition of a curve that I know of is "a function from an interval to the space/manifold", and then you can identify curves which differ only by reparametrization. So it seems like you need a new definition of a curve if you want to avoid parametrizations.

 

[mma]

I only see an alternative definition of the distance on a Riemannian manifold, not of the arc-length of a curve, am I missing something?

Yes, the definition of the arc length isn't given here. It is as usual, the limit of the sum of the distances of finite numbers of successive points on the curve as the maximum of these distances approaches to 0.

Also, the only definition of a curve that I know of is "a function from an interval to the space/manifold", and then you can identify curves which differ only by reparametrization. So it seems like you need a new definition of a curve if you want to avoid parametrizations.

I mean the images of curves.

 

[yyat]

Yes, the definition of the arc length isn't given here. It is as usual, the limit of the sum of the distances of finite numbers of successive points on the curve as the maximum of these distances approaches to 0.

I mean the images of curves.

In both of these statements you are assuming (I think) that the image of the curve is an embedded manifold, but a general curve can be non-injective and non-immersive.
For example, how do you define successive points without a parametrization? Think for example of the curve that goes around the unit circle a few times, stopping and chaning direction in the process.

 

[mma]

In both of these statements you are assuming (I think) that the image of the curve is an embedded manifold, but a general curve can be non-injective and non-immersive.
For example, how do you define successive points without a parametrization? Think for example of the curve that goes around the unit circle a few times, stopping and chaning direction in the process.

OK. I mean one-dimensional connected, simply connected submanifolds.

 

[yyat]

OK. I mean one-dimensional connected, simply connected submanifolds.

In that case the path-integral is the same as the usual integral of a 1-form on a 1-D manifold, but that is defined in terms of charts (is there a different definition?), which are (local) parametrizations.

I think you need to explain why you want to get rid of parametrizations in the path integral. For instance, the alternative definition of the distance you gave is useful in spectral geometry because the other one doesn't work. I am not sure how integrals are defined in spectral geometry, but maybe the answer to your question lies there.

 

[mma]

I think you need to explain why you want to get rid of parametrizations in the path integral.

I have only aesthetic reasons and curiosity.

Unfortunately, I don't know spectral geometry.

 

[yyat]

I have only aesthetic reasons and curiosity.

Unfortunately, I don't know spectral geometry.

You might want to check out Chapter 6.1 of http://alainconnes.org/en/downloads.php" by A. Connes. It also talks about the alternative definition of the distance you mentioned and gives a definition of the integral in terms of a trace.

 

[mma]

You might want to check out Chapter 6.1 of http://alainconnes.org/en/downloads.php" by A. Connes. It also talks about the alternative definition of the distance you mentioned and gives a definition of the integral in terms of a trace.

Thank you for the link to this excellent book. But I'm afraid it's of little avail to me.