Proving the first three fundamental forms for Differential Geometry

Asked May 07 '19 at 16:26

Active May 07 '19 at 16:26

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I need to write w=ae₁+be₂ in terms of the basis {e₁,e₂} of principal directions at p. Further, I need to compute the three fundamental forms at w, that is Iₚ(w),IIₚ(w), IIIₚ(w) while simplifying expressions using the face that e₁,e₂ are principal directions.

Lastly, I need to show that three fundamental forms satisfy the following

IIIₚ(w)-2H(IIₚ(w))+ KIₚ(w)=0

Any help on this would be greatly appreciated.

asked May 07 '19 at 16:26

JohnFreeman

3

Your question is good, but you are going to get a ton of downvotes if you do not show some work. What have you tried? Also, please post using MathJax or a like program! Welcome to Mathstack- – Matematleta May 07 '19 at 16:29
What is H, and what is K? – D. Brito May 07 '19 at 20:00
@D.Brito $H$ is the mean curvature, and $K$ is the Gaussian curvature of the surface – Yuri Vyatkin May 07 '19 at 23:35
Basically, this is a consequence of the Cayley-Hamilton theorem. Please read my answer. – Yuri Vyatkin May 07 '19 at 23:39
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, use MathJax. – dantopa May 20 '19 at 01:24

Proving the first three fundamental forms for Differential Geometry

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