# Are there operators that change the curvature of manifolds?

#### BWV

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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#### Tac-Tics

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

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

#### Tac-Tics

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.

#### wofsy

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.

#### Tac-Tics

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.

#### wofsy

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

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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.

#### zhentil

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.

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