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Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I have not been able to find it.

Alternatively, how far can one get into the theory of surfaces without introducing coordinate charts?

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    $\begingroup$ You can define the Levi-Civita connection, the first and the second fundamental form without appealing to local coordinates. You can even prove Gauss-Bonnet without local coordinates but this assuming some topological facts: degree theory and Poincaré-Hopf theorem $\endgroup$ Dec 13, 2018 at 21:35
  • $\begingroup$ What subjects do you consider classical? The book "Differential Topology" by Guillemin and Pollack covers quite a lot of classical ground. I have some notes that extend G&P into basic differential geometry, using the same general framework as G&P. $\endgroup$ Dec 13, 2018 at 22:25

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Maybe you are thinking of the kind of work that people such as Jamie Sethian do. They use level set methods to study the evolution of surfaces under various PDE in which the topology of the surface can change, usually because one passes through a critical point of the function whose level sets give the evolving surface. Try looking at his book Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science.

Also, while I'm not aware of a book, per se, geometric measure theorists do tend to use level set methods in the study of mean curvature flow and inverse mean curvature flow. You might take a look a the papers of Brian White and Tom Ilmanen in particular.

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