# Theory of surfaces in $\mathbb{R}^3$ as level sets

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?

• 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 Dec 13, 2018 at 21:35
• 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. Dec 13, 2018 at 22:25