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Hi all! I just found this site today, and I am really hoping that I can get some useful advice here. That said, I have two problems--one easy, one not so easy.

Easy problem:

Basically, I was wondering if anybody out there knows of an algorithm to calculate [itex] g_{ij} [/itex], given only the 'shape' of the underlying manifold? For example, I have what can best be described as a dumbell, or [itex] S^2 [/itex] with the equator contracted in to a small radius--no corners anywhere, though. Keep in mind that I am not looking so much for a solution for the dumbell, but something that could be applied to a more general, finite dimensional, connected [itex] C^\infty [/itex] manifold.

Harder problem:

As a related problem, I am also trying to come up with some sort of meaningful geometric interpretation of the Laplacian when it is being applied to some function that lies on a Riemannian manifold, rather than Euclidean space. In particular, it seems that something like the heat equation:

[itex]\frac{\partial}{\partial t}u=\Delta u [/itex]

should lend itself to a, let's say, rewording as something like a Gauss curvature flow:

[itex]\frac{\partial}{\partial t}x=-K \nu [/itex]

where [itex] x [/itex] is an embedding. Of course, the manifold where this new surface would be embedded would require some additional structure, since the geometric heat equation is valid only for (smooth) compact Riemannian surfaces without boundary--like [itex] S^2 [/itex], or something diffeomorphic to it. That problem, however, has already been taken care of, so please don't waste your time even considering it.

Anyway, I thank you all in advance for your time, and if you require clarification of the question at all, please don't hesitate to ask.

Cheers!

Easy problem:

Basically, I was wondering if anybody out there knows of an algorithm to calculate [itex] g_{ij} [/itex], given only the 'shape' of the underlying manifold? For example, I have what can best be described as a dumbell, or [itex] S^2 [/itex] with the equator contracted in to a small radius--no corners anywhere, though. Keep in mind that I am not looking so much for a solution for the dumbell, but something that could be applied to a more general, finite dimensional, connected [itex] C^\infty [/itex] manifold.

Harder problem:

As a related problem, I am also trying to come up with some sort of meaningful geometric interpretation of the Laplacian when it is being applied to some function that lies on a Riemannian manifold, rather than Euclidean space. In particular, it seems that something like the heat equation:

[itex]\frac{\partial}{\partial t}u=\Delta u [/itex]

should lend itself to a, let's say, rewording as something like a Gauss curvature flow:

[itex]\frac{\partial}{\partial t}x=-K \nu [/itex]

where [itex] x [/itex] is an embedding. Of course, the manifold where this new surface would be embedded would require some additional structure, since the geometric heat equation is valid only for (smooth) compact Riemannian surfaces without boundary--like [itex] S^2 [/itex], or something diffeomorphic to it. That problem, however, has already been taken care of, so please don't waste your time even considering it.

Anyway, I thank you all in advance for your time, and if you require clarification of the question at all, please don't hesitate to ask.

Cheers!

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