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- Using the principal curvatures of an embedded surface of positive Gauss curvature, what is the curvature of the derived metric?

Start with a closed surface of positive Gauss curvature embedded smoothly in ##R^3##. At each point, choose two independent eigenvectors of the shape operator whose lengths are the corresponding principal curvatures. By declaring them to be orthonormal one gets - I think - a new metric on the surface.

- How does one compute the Gauss curvature of this new metric?

- Can the surface also be embedded in ##R^3##?

- Does this metric also have positive Gauss curvature? If so repeat the process and calculate the Gauss curvature again. In case one gets an infinite sequence of surfaces of positive Gauss curvature what is the limiting metric?

- Does one always get a limiting metric?

- How does one compute the Gauss curvature of this new metric?

- Can the surface also be embedded in ##R^3##?

- Does this metric also have positive Gauss curvature? If so repeat the process and calculate the Gauss curvature again. In case one gets an infinite sequence of surfaces of positive Gauss curvature what is the limiting metric?

- Does one always get a limiting metric?

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