show that $K=E_2[\omega_{12}(E_1)]-E_1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$?

Question

$$K=E_2[\omega_{12}(E_1)]-E_1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$$

where $K$ is Gaussian curvature, $E_i$'s are tangent frame field on surface $M$ in $R^3$, $v[\cdot ]$ is directional derivative in $v$ direction, $\omega_{ij}$ is connection form.

my textbook gives a hint that write $\omega_{12}=f_1\theta_1+f_2\theta_2$, where $f_i=\omega_{12}(E_i)$, $\theta_i$ is dual form, and use $$d\omega_{12}=-K\theta_1\wedge \theta_2,$$ where $\wedge $ is wedge product. (O'Neil, section 6.3 exercise 2-(a))

I'm having trouble with how to take exterior derivative of $\omega_{12}$, confused if $f_i$ is 1-form or not(looks not since dual forms are 1-form)

Answer 1

The $f_i$ are, indeed, scalar functions. You need three ingredients:

• the product rule
• the formulas $(df_1\wedge\theta_1)(E_1,E_2)=-df_1(E_2)$, etc.
• the structure equations for $d\theta_i$.