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I am trying to understand geometric flows, and in particular the Ricci flow. I understand how to calculate the metric tensor from the parametrization of a surface, but I am facing a problems in the concept phase.

A metric tensor's purpose is to provide a coordinate invariant expression for the shape (in the form of distance) of an object, i.e. a surface. A geometric flow changes the metric tensor. Therefore, the shape of the manifold will change, obviously for any coordinate system.

So, what exactly is the product of a geometric flow? Let's stick to surfaces for now, so is it the original surface endowed with a new measure of distance? The way I understand it is that since the shape of the original surface is invariant, changing how we measure distance on the surface shouldn't yield a different shape. So does this mean that by defining a new metric for a surface we get a different surface?

If so, I am missing something here because I can't understand how defining a new measure of distance for an object creates a new, different object.

I would be grateful if someone could showcase or guide me through applying a geometric flow for a simple shape, i.e. an ellipsoid.