I am trying to understand the geometry of curves and surfaces, in
particular, surfaces that are obtained through the motion curves.
Let us consider the space curve $\gamma :I\rightarrow $ $\mathbb{R}^{3}$
such that $\gamma =\gamma \left( s,t\right) ,$ where $s$ is an arc-length
parametrization and $t$ is another parametrization. As is known, we can now
consider $\gamma \left( s,t\right) $ as a swept surface and the normal of
the surface is given by
\begin{equation}
\mathbf{N=}\frac{\gamma _{s}\times \gamma _{t}}{\left\Vert \gamma _{s}\times
\gamma _{t}\right\Vert }.
\end{equation}
We can also find the coefficients of the first and second fundamental forms
due to the following identities
\begin{eqnarray}
I &=&\left\langle \gamma _{s}ds+\gamma _{t}dt,\gamma _{s}ds+\gamma
_{t}dt\right\rangle , \\
II &=&\left\langle \gamma _{s}ds+ \gamma _{t} dt,N _{s}ds+N_{t}dt\right\rangle .
\end{eqnarray}
As an example, say $\gamma _{t}=\gamma _{s}\times \gamma _{ss}=\kappa b,$
which is known as the binormal motion of curves. Here, $t$ is a tangent vector, $%
b$ is a binormal vector, and $\kappa $ is a curvature of the curve. There also
exists a normal vector $n$ along with the curve $\gamma .$ Thus, $\gamma
\left( s,t\right) $ is a surface satisfying the following
\begin{equation}
\mathbf{N}_{\gamma }=\frac{t\times \kappa b}{\left\Vert t\times \kappa
b\right\Vert }=-n,\ t\times b=-n.
\end{equation}
As a result, one can figure out the coefficients of the first and second
fundamental forms and related concepts due to Eqs. $\left( 1-3\right) $. My
question is the following. Can we just use Eqs. $\left( 1-3\right) $ to
derive the surface normal and related information for any vector field that
is assumed to move or do we need to manipulate the formulas? Let me make it
more clear. Let's say we have the following vector field
\begin{equation}
\mathbf{V}=\kappa t+\tau n.
\end{equation}
Can I just move the vector field just like the curve and finally get a new
swept surface $\mathbf{V=V}\left( s,t\right) $ whose surface normal is
written by
\begin{equation}
\mathbf{N}_{V}=\frac{\mathbf{V}_{s}\times \mathbf{V}_{t}}{\left\Vert \mathbf{%
V}_{s}\times \mathbf{V}_{t}\right\Vert }.
\end{equation}
Thanks in advance.
