# How is the shape operator related to the second fundamental form?

I don't understand how the shape operator $\Theta$ is related to the second fundamental form $\Pi$.

Before now, we have derived the second fundamental form as a quadratic form with the matrix $$\begin{pmatrix} L & M \\ M & N \end{pmatrix}.$$

I understand the shape operator as a map $\Theta : T_p (S) \to T_p(S)$ where we get that $\Theta (e) = -D_e \, n$ which is the derivative of $n$ in the direction of $e$.

Now my lecture notes state that: in these terms, we can write the second fundamental form as $$\Pi(e,f) = e.\Theta (f) = -e.D_f \, n$$

Where has this 'dot product' definition of the second fundamental form come from?

Thanks!

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Can you say a bit more about what $L$, $M$ and $N$ are in your definition of the second fundamental form? In my mind, the second formulation you have is the definition, so I'd like a clearer idea of what your definition is. –  Sam Lisi Mar 24 '13 at 17:35