Are there operators that change the curvature of manifolds?

  • Thread starter BWV
  • Start date

BWV

476
388
Seen a couple of pieces on gauge theories of finance equating arbitrage opportunities to curvature, for this to work the curvature must therefore not be constant, instead it would be some sort of a function of capital flows
 
809
6
Seen a couple of pieces on gauge theories of finance equating arbitrage opportunities to curvature, for this to work the curvature must therefore not be constant, instead it would be some sort of a function of capital flows
Are the manifolds you're working with codify the time dimension? If the curvature is changing with respect to time, then so must your manifold. But the curvature should always be constant with respect to the manifold at any point in time.
 

BWV

476
388
Are the manifolds you're working with codify the time dimension? If the curvature is changing with respect to time, then so must your manifold. But the curvature should always be constant with respect to the manifold at any point in time.
The curvature would change over time, but not be dependent on the time variable - other variables, perhaps stochastic, would determine the curvature
 
809
6
My point is simply that if you're talking about gaussian curvature, it IS constant with respect to the manifold. If your curvature is changing, then so is your manifold.
 
709
0
My point is simply that if you're talking about gaussian curvature, it IS constant with respect to the manifold. If your curvature is changing, then so is your manifold.
It is possible to imagine a manifold in which its geometry is changing in time - e.g. using a differential equation.
 
809
6
It is possible to imagine a manifold in which its geometry is changing in time - e.g. using a differential equation.
But then you're not dealing with a constant manifold. You're dealing with a manifold-valued function. In such a case, all properties which depend on the manifold transitively depend on the time parameter as well.
 
709
0
But then you're not dealing with a constant manifold. You're dealing with a manifold-valued function. In such a case, all properties which depend on the manifold transitively depend on the time parameter as well.
The Riemannian manifold is not constant but the differentiable manifold is.
 

quasar987

Science Advisor
Homework Helper
Gold Member
4,771
7
It is possible to imagine a manifold in which its geometry is changing in time - e.g. using a differential equation.
I think that an example of such a D.E. is the famous Ricci flow, which plays an important role in the solution of the Poincaré conjecture by Perelman: http://en.wikipedia.org/wiki/Ricci_flow.
 
489
0
A Riemannian metric is an easily perturbed object. The discussion above is semantics, of course (if we're talking about the curvature of a manifold, it's necessary Riemannian, and if we change the metric, it's a different Riemannian manifold). The only thing that has to be preserved is the integral of the curvature, if the manifold is compact. I imagine that any reasonable financial theory would assume this, however.
 

Related Threads for: Are there operators that change the curvature of manifolds?

  • Posted
Replies
1
Views
704
Replies
31
Views
3K
Replies
8
Views
3K
  • Posted
Replies
8
Views
645
Replies
2
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top