I have a question regarding fractional calculus, namely, what is the line element "$$ds^2 = g^{\mu\nu} dx_\mu dx_\nu$$ in fractional "n" dimensions?
I am aware of some formulas that give the surface of a sphere in general (fractional) dimensions $$S(n) = \frac{2\pi^\frac{n}{2}}{\Gamma(\frac{n}{2})} $$, also the different type of Differintegral formulas (such as the Riemann-Liouville formula): $$\int_a^b dx^\alpha f(x) = \frac{1}{\Gamma(\alpha)}\int_a^b (b-t)^{\alpha-1}f(t)dt$$.
Along this line, what is the function of the sphere? Should it be related maybe to the norm of a fractional dimensional vector?
In general I'm approaching the problem $$\int_a^b dx^{\alpha(x)} \sqrt{g(x)}f(x)$$, where $$\alpha(x)$$ is a "scale dependent" dimension factor, $$g(x)$$ should be the determinant of the metric tensor and $$f(x)$$ is a general function, and of course both $$g(x)=g_\alpha(x),f(x)=f_\alpha(x)$$ should be aware of the changing of $$\alpha(x)$$, thus I'm looking for a "fractional metric tensor". I couldn't really find much about this topic. I found some approaches to multifractional calculus, but so far nothing that helps with my main question.