Category Archives: Classical Differential Geometry

Definition. The dual 1-forms $\theta_1,\theta_2,\theta_3$ of a frame $E_1,E_2,E_3$ on $\mathbb{E}^3$ are defined by $$\theta_i(v)=v\cdot E_i(p),\ v\in T_p\mathbb{E}^3.$$ Clearly $\theta_i$ is linear. Example. The dual 1-forms of the natural frame $U_1,U_2,U_3$ are $dx_1$, $dx_2$, $dx_3$ since $$dx_i(v)=v_i=v\cdot U_i(p)$$ for each … Continue reading →

Let $E_1, E_2, E_3$ be an arbitrary frame field on $\mathbb{E}^3$. At each $v\in T_p\mathbb{E}^3$, $\nabla_v E_i\in T_p\mathbb{E}^3$, $i=1,2,3$. So, there exists uniquely 1-forms $\omega_{ij}:T_p\mathbb{E}^3\longrightarrow\mathbb{R}$, $i,j=1,2,3$ such that \begin{align*} \nabla_vE_1&=\omega_{11}(v)E_1(p)+\omega_{12}(v)E_2(p)+\omega_{13}(v)E_3(p),\\ \nabla_vE_2&=\omega_{21}(v)E_1(p)+\omega_{22}(v)E_2(p)+\omega_{23}(v)E_3(p),\\ \nabla_vE_3&=\omega_{31}(v)E_1(p)+\omega_{32}(v)E_2(p)+\omega_{33}(v)E_3(p) \end{align*} for each $v\in T_p\mathbb{E}^3$. These equations are … Continue reading →

In Euclidean 3-space $\mathbb{E}^3$, we have naturally defined frame $U_1(p)$, $U_2(p)$, $U_3(p)$ for each $p\in\mathbb{E}^3$, where $U_1=(1,0,0)$, $U_2(0,1,0)$, $U_3=(0,0,1)$. The frame $U_1$, $U_2$, $U_3$ (as vector fields) is called the natural frame. As a generalization of the natural frame, we … Continue reading →