## Second fundamental form of hypersurface

Suppose is an embedding map. is an open set. Consider the Gauss equation of hypersurface

,

is the outer normal vector. The Codazzi equation

Then the second fundamental form of is , where

There are two ways to express the second fundamental form, one is from , the other is from . **Firstly**, let us consider from

Extend and to a vector field and on

,

Note the fact that , we get

**Secondly,** let us consider from . Using the same extension process, one can prove

,

So is equivalent to

see the notation on https://sunlimingbit.wordpress.com/2013/02/10/tangential-gradient-on-the-hypersurface/

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