@JohnB Thank you! are there some dynamical interpretations for this spectrum condition?

I am realy curious abiut the reason you impose the condition "The preimage of $\infty$ is nowhere dense" when we are sure that the range of $f$ does not approach infinity

Moreover since the algebra $C(X)$ is a commutative model of a von Neumann algebra, I googled "Von neumann algebra + RH" I got some things relevant

I am talking about $C^+(X)$ definition

How is it possible $f$ take value $\infty$ .?You assumed that $X$ is compact so $f(X)$ is far from the north pole,

@BenMcKay Sorry I revise the latex of my previous 2 comment: Now with some abuse of notation and abuse of terminologies one arrive at $\omega \wedge \omega \wedge \omega=-\omega \wedge d\omega =0???$

$(???)=0$ in case of integrability. By the way did you discuss this kind of integrability in your lecture?(The integrability after composition with one(hence all) functionals defined on $T_e G$?

@BenMcKay I wonder if the nth power is always non zero? Some intuitions: Let's pose the question of integrability of Cartan form in the following sense: Choose a functional $\phi$on the tangent space(the Lie algebra) then $\phi \circ \omega$ is a 1 form in the usual sense hence integrability of $\phi\circ \omega$ is in the hand of Frobenius condition $\alpha \wedge d\alpha=0$ Now with some abuse of notation and abuse of terminologies one arrive at $\wmega \wedge \omega \wedge \omega=-\omega \wedge d\omega= 0???)

@BenMcKay Thank you Ben for sharing your excelent lecture: I was revewing its furst parts, Cartan form $\omega$ as a lie algebra valued form. Is it an obvious question to ask about the integral $\int_G \omega^n$ as an element of the Lie algebra?What is this integral?

@BenMcKay I would appreciate if you provid the link of your note on Cartan Geometries.

@BenMcKay I wonder under what conditions the group $Ric_g$ is a finite dimensional Lie group? and what is a sharp upper bound for its dimension(Inspired by similar result on isometric group). Any way thank youn very much for your attention to the question

@BenMcKay what about finit index case:under what condition the isometry groyp is a finit3 index subgroup?