Are there operators that change the curvature of manifolds?

In summary, there have been discussions on gauge theories of finance that relate arbitrage opportunities to curvature. However, for this to be valid, the curvature must vary and not be constant. It could be dependent on factors such as capital flows and may change over time. This would result in a manifold-valued function rather than a constant manifold. An example of this is the Ricci flow, which is often used in solving mathematical problems such as the Poincaré conjecture. Despite changes in the geometry, the integral of the curvature should remain constant if the manifold is compact, which is typically assumed in financial theories.
  • #1
BWV
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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
 
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  • #2
BWV said:
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.
 
  • #3
Tac-Tics said:
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
 
  • #4
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.
 
  • #5
Tac-Tics said:
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.
 
  • #6
wofsy said:
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.
 
  • #7
Tac-Tics said:
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.
 
  • #8
wofsy said:
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.
 
  • #9
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.
 

1. What are operators that change the curvature of manifolds?

The most commonly known operators that change the curvature of manifolds are the Laplace-Beltrami operator and the Ricci flow operator. These operators are used in differential geometry and are essential in studying the geometry of curved spaces.

2. How do these operators change the curvature of manifolds?

The Laplace-Beltrami operator acts on functions defined on the manifold and changes their curvature by transforming their values at each point. The Ricci flow operator, on the other hand, changes the metric tensor of the manifold by evolving it in a way that minimizes the Ricci curvature.

3. Can these operators change the curvature of any type of manifold?

Yes, these operators can be applied to any Riemannian manifold, which is a type of manifold that has a well-defined metric tensor. This includes manifolds with positive, negative, or zero curvature.

4. What is the significance of studying these operators?

Studying these operators allows us to understand the geometry of curved spaces and how they can be deformed or transformed. This has important applications in fields such as physics, where the curvature of spacetime is related to the distribution of matter and energy.

5. Are there any real-life examples of manifolds whose curvature can be changed by these operators?

Yes, one example is the sphere, which can be transformed into a cylinder or a cone by applying the Laplace-Beltrami operator. Another example is the Ricci flow, which has been used to study the formation of black holes in astrophysics.

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