Idea
Let's try to express objects in terms of their line elements and line element vector rather than the co-ordinate system. Let the object be a line in three dimensions:
Line
At the origin we have: $$ d \vec s|_{(0,0,0)} = dx \hat i + dy \hat j + dz \hat k $$
$$ d \vec s|_{(a,a,a)} = dx \hat i + dy \hat j + dz \hat k $$
Hence the equation of this line in terms of it's line element vector is:
$$ d \vec s|_{(0,0,0)} = d \vec s|_{(a,a,a)} $$
This can be easily the equation of a line in $n$ dimensions passing through the origin!
Hence, the equation of a line is:
$$ d \vec s|_{\vec r} = d \vec s|_{\vec r + \vec \epsilon}$$
Shell
To find the equation of the surface of a unit sphere:
Let the co-ordinate system be:
$$ d \vec s |_{\theta,\phi} = d \theta \hat \theta + sin \theta d \phi \hat \phi $$ $$ d \vec s |_{\theta+ \epsilon ,\phi } = d \theta \hat \theta + sin (\theta +\epsilon )d \phi \hat \phi $$
Then the inner product of both line elements are:
$$d \vec s |_{\theta,\phi} \cdot d \vec s |_{\theta + \epsilon,\phi } = d \theta^2 + sin(\theta) sin(\epsilon + \theta ) d \phi^2 $$
$$ = d \theta^2 + sin^2 (\theta) cos(\epsilon) d \phi^2 + cos(\theta)sin(\epsilon) d \phi^2 $$
Keeping $1$'st order terms in $\epsilon$:
$$ = d \theta^2 + sin^2 (\theta) d \phi^2 + cos(\theta) \epsilon d \phi^2 $$
But this is nothing more than:
$$ = |d \vec s |_{\theta,\phi}|^2 + (\lim_{\epsilon \to 0} \frac{d \vec s |_{\theta + \epsilon,\phi} - d \vec s |_{\theta,\phi}}{\epsilon}) \cdot ( \epsilon \hat \phi d \phi) $$
Hence, the general equation of a shell is (?):
$$ \lim_{\epsilon \to 0} \frac{d \vec s |_{\theta,\phi, \dots, \kappa} \cdot d \vec s |_{\theta +\epsilon,\phi,\dots, \kappa } - |d \vec s |_{\theta,\phi, \dots, \kappa}^2}{\epsilon} = (\lim_{\epsilon \to 0} \frac{d \vec s |_{\theta +\epsilon,\phi,\dots, \kappa } - d \vec s |_{\theta ,\phi,\dots, \kappa }}{\epsilon}) \cdot ( \hat \phi d \phi) $$
This is the same as:
$$ (\lim_{\epsilon \to 0} \frac{d \vec s |_{\theta +\epsilon,\phi,\dots, \kappa } - d \vec s |_{\theta ,\phi,\dots, \kappa }}{\epsilon}) \cdot (d \vec s|_{\theta +\epsilon,\phi,\dots, \kappa } - ( \hat \phi d \phi ))= 0 $$
where $d \vec s |_{\theta +\epsilon,\phi,\dots, \kappa } - ( \hat \phi d \phi)) \neq \vec 0 $
Verification of shell in different coordinate system:
The equation of a unit shell in $3$-D is: $$ x^2 + y^2 + z^2 = 1 $$ $$\implies dz =-\frac{x dx + ydy}{\sqrt{x^2 + y^2 -1}}$$
Hence, the line element is: $$ ds|_{(x,y,z)} = dx \hat i + dy \hat j -\frac{x dx + ydy}{\sqrt{x^2 + y^2 -1}} \hat k $$ Similarily, the line element at a different point is: $$ ds|_{(x+ \epsilon,y,z)} = dx \hat i + dy \hat j -(\frac{x dx + ydy}{\sqrt{(x + \epsilon)^2 + y^2 -1}} + \frac{ \epsilon dx }{\sqrt{(x+ \epsilon)^2 + y^2 -1}}) \hat k $$
Expanding to first order in $\epsilon$ we get: $$ ds|_{(x+ \epsilon,y,z)} = dx \hat i + dy \hat j -(\frac{x dx + ydy}{\sqrt{(x )^2 + y^2 -1}}- \frac{ \epsilon x (x dx + ydy)}{{(x^2 + y^2 -1)}^{3/2}} + \frac{ \epsilon dx }{\sqrt{(x)^2 + y^2 -1}}) \hat k $$
Simplifying a bit we get:
$$ ds|_{(x+ \epsilon,y,z)} = dx \hat i + dy \hat j -(\frac{x dx + ydy}{\sqrt{(x )^2 + y^2 -1}}- \frac{ \epsilon x (x dx + ydy)}{{(x^2 + y^2 -1)}^{3/2}} + \frac{ \epsilon {(x^2 + y^2 -1)}dx }{{(x^2 + y^2 -1)}^{3/2}}) \hat k $$
Further simplyfying:
$$ ds|_{(x+ \epsilon,y,z)} = dx \hat i + dy \hat j -(\frac{x dx + ydy}{\sqrt{(x )^2 + y^2 -1}}- \frac{ \epsilon ( xydy)}{{(y^2 -1)}^{3/2}} + \frac{ \epsilon {( y^2 -1)}dx }{{(x^2 + y^2 -1)}^{3/2}}) \hat k $$
Hence,
$$ ds|_{(x+ \epsilon,y,z)} \cdot ds|_{(x,y,z)} = dx^2 + dy^2 +(\frac{x dx + ydy}{\sqrt{(x )^2 + y^2 -1}}) (\frac{x dx + ydy}{\sqrt{(x )^2 + y^2 -1}}- \frac{ \epsilon ( xydy)}{{(x^2 + y^2 -1)}^{3/2}} + \frac{ \epsilon {( y^2 -1)}dx }{{(x^2 + y^2 -1)}^{3/2}}) $$
Hence,
$$ ds|_{(x+ \epsilon,y,z)} \cdot ds|_{(x,y,z)} = dx^2 + dy^2 +(\frac{x dx + ydy}{\sqrt{(x )^2 + y^2 -1}})^2 -(\frac{x dx + ydy}{\sqrt{(x )^2 + y^2 -1}}) (\frac{ \epsilon ( xydy)}{{(x^2 + y^2 -1)}^{3/2}} + \frac{ \epsilon {( y^2 -1)}dx }{{(x^2 + y^2 -1)}^{3/2}}) $$
Which is equal to:
$$ = |d \vec s |_{\theta,\phi}|^2 + (\lim_{\epsilon \to 0} \frac{d \vec s |_{x + \epsilon,y,z} - d \vec s |_{x,y,z}}{\epsilon}) \cdot ( \epsilon \hat k d k) $$
Questions
What is the subbranch of differential geometry known as? (studying shapes in this fashion?) Is the general equation of a shell correct? (If not, please show the correct expression?) How to show uniqueness of solutions? Will this idea work if one thinks of it as an embedding in a higher Euclidean dimension? (If yes, why wont it work without that)? Can any shape be expressed in terms of line element (I think the answer should be yes but I'm not sure how to prove it)?