The general expression of the line element $ds^2$ is $$ds^2 = g_{ij}dX^{i}dX^{j},$$ where $g_{ij}$ is an element of the metric tensor.
Is there a rigorous proof of why there are no terms in the expression of $ds^2$ that are of higher order than $2$? Would it be possible to construct a space where this would be true (even if this space could not be used in physics)?
