Consider the unit sphere in $\mathbb R^3$, let $\theta$ be polar angle, $\varphi$ be azimuthal angle. Then, the standard metric is $$ g= \begin{pmatrix} 1 & 0 \\ 0 & \sin^2\theta \end{pmatrix} $$ I try to change the metric, for example, let $$ \hat g = \begin{pmatrix} 1 &0 \\ 0 &(\theta^2+1)\sin^2\theta \end{pmatrix} $$ I can't image the shape of $(S^2,\hat g)$. So, I want some software which can give the sharp of given metric on $S^2$. Namely, the sharp of isometric imbedding of $(S^2,\hat g)$ into $\mathbb R^3$.
In fact, I want to see the element of metric how to change the sharp.
Maybe, there is no such software. If so, there is any algorithm to realize it ? I know a little Python.
You need more information for $2$ dimensional surfaces: given the two symmetric bilinear forms $g(x),h(x)$ ($x\in U\subset\mathbb{R}^2$ comes from the parametrization) which are $C^2$ and $C^1$, satisfying the Codazzi-Mainardi and the Gauss equation, there exists a unique (up to translation and rotation) surface such that $g$ is the first fundamental form and $g$ is the second fundamental form of this surface. This is the statement of the fundamental theorem of surfaces. Looking at the proof of this theorem, one sees that you only need to solve a system of linear differential equations. If you numerically solve these equations for given $g,h$, you get your desired surface.