# Handling different kind of indices in single tensor equation

I want to solve the following differential equation,

$$\partial_a (\sqrt{h} \, h^{a b} \partial_b X^{i}) = 0$$

where $$X^i \equiv \{X^1(\sigma^1, \sigma^2), X^2(\sigma^1, \sigma^2), X^3(\sigma^1, \sigma^2) \},$$ $$\partial_a := \frac{\partial}{\partial \sigma^a},$$ $$h_{a b} \equiv \frac{\partial X^i}{\partial \sigma^a} \frac{\partial X^j}{\partial \sigma^a} \, \delta_{i j},$$ $$h := det(h_{ab})$$

Note that here $$a,b$$ indices run from 1 to 2, whereas the $$i, j$$ indices run from 1 to 3. I am struggling to write down the differential equation because I am not able to properly contract the $$a, b$$ indices.

Here is a code where I defined the quantities but failed to write down the equation.

         \[DoubleStruckCapitalX] = Table[X[i][\[Sigma][1], \[Sigma][2]], {i, 1, 3}];
\[DoubleStruckS] = Table[\[Sigma][a], {a, 1, 2}];
h[i_, j_] := D[\[DoubleStruckCapitalX], \[DoubleStruckS][[i]]] . D[\[DoubleStruckCapitalX], \[DoubleStruckS][[j]]]
\[DoubleStruckH] = ( {{h[1, 1], h[1, 2]}, {h[2, 1], h[2, 2]}} ) // FullSimplify;
Inverse[\[DoubleStruckH]] . \[DoubleStruckH] // FullSimplify // MatrixForm
Sqrt[Det[\[DoubleStruckH]]] // FullSimplify


Changing some symbol names for clarity of reading:

S = Table[Subscript[\[Sigma], a], {a, 1, 2}];
X = Table[Apply[Subscript[x, i], S], {i, 1, 3}];

div[V_, S_, H_] := TensorContract[Grad[Sqrt[Det[H]] V . Inverse[H], S], ArrayDepth[V] + {0, 1}]

div[Grad[X, S], S, H] // Simplify