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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
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.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
The curvature would change over time, but not be dependent on the time variable - other variables, perhaps stochastic, would determine the curvatureAre 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.
It is possible to imagine a manifold in which its geometry is changing in time - e.g. using a differential equation.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.
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.It is possible to imagine a manifold in which its geometry is changing in time - e.g. using a differential equation.
The Riemannian manifold is not constant but the differentiable manifold is.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.
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.It is possible to imagine a manifold in which its geometry is changing in time - e.g. using a differential equation.