Below are the slides from our lectures on smooth surfaces, with a bit of reorganization and minor additions. After wrapping up smooth surfaces we’ll take a look at the discrete picture.

## 2 thoughts on “Slides — Surfaces”

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The motivation for S (the shape operator) is something like this :

“The change in N at any given point has to be normal to N (since N is a unit vector field), so dN must always be tangent to the surface. So, there must be some linear map S such that dN = df(SX)”. I have two basic-ish questions :

1) This tacitly assumes that df is surjective (as a map from vector fields to vector fields), i.e. any tangent to the surface (at point p) must be in the range of df (at p). Is this true for smooth surfaces in general?

2) Assuming that df really is surjective, we can say that there is some map h such that dN = df(h(X)), but how do we conclude that this map has to be linear?

Could you post the slides from Thursday? I had to miss class and would like to catch up before the next lecture on Tuesday. Thanks.