Consider the functions $x^2$ and $x^4 + 2x^2y^2$ on the unit sphere $S^2$. The surface integral of these functions over the sphere can easily be calculated by symmetry via $$3 \iint_{S^2} x^2 \mathrm{d}A = \iint_{S^2} (x^2 + y^2 + z^2) \, \mathrm{d}A = \iint_{S^2} \mathrm{d}A = 4\pi$$ and $$3 \iint_{S^2} (x^4+2x^2y^2)\, \mathrm{d}A = \iint_{S^2} (x^2 + y^2 + z^2)^2 \, \mathrm{d}A = \iint_{S^2} \mathrm{d}A = 4\pi.$$

However, I suspect (although I cannot prove) that the function $x^4$ cannot be integrated without direct parameterization of the sphere and evaluation of the surface integral.

My question is: in general, given any symmetries and polynomial relations on a manifold (in this case $(x, y, z) \mapsto (y, z, x)$ and $x^2 + y^2 + z^2 = 1$), is there a general theory to determine what functions are integrable over the manifold by symmetry and relations alone?

A reference (or definitive statement of lack thereof) would be greatly appreciated.

  • $\begingroup$ These are discussions likely present in several different books on Lie Theory. Groups are a way to study symmetries of Mathematical objects, and Lie Groups are groups with a smooth manifold structure. $\endgroup$ – user458276 Mar 15 '19 at 21:31
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    $\begingroup$ The only allowed operations are symmetrizing and using the fact that $x^2 + y^2 + z^2$, so the condition might just be that the symmetrization $\mathfrak{S}(f)$ of the integrand $f$ (in $x, y, z$) must by a polynomial in $x^2 + y^2 + z^2$. For $n > 1$ the power sum monomial $\mathfrak{S}(x^{2n}) = x^{2n} + y^{2n} + z^{2n}$ is not a power of $x^2 + y^2 + z^2$, so no integral $\iint_{S^2} x^{2n} dA$ can be evaluated with this trick. $\endgroup$ – Travis Willse Mar 16 '19 at 3:35
  • $\begingroup$ This in particular implies that the only homogeneous polynomials $f$ of degree $2n$ to which the trick applies are those whose symmetrization $\mathcal{S}(f)$ is a constant multiple of $(x^2 + y^2 + z^2)^n$. $\endgroup$ – Travis Willse Mar 16 '19 at 3:38

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