Let $M$ be a finite dimensional smooth manifold endowed with a riemannian metric $g$ and a smooth action $\mu$ by a compact Lie group $G$. Averaging $g$ over $G$ defines a new metric $$g'(X,Y)=\int_Gg(d\mu_aX,d\mu_aY)da,$$ (integral with respect to a invariant Haar measure) for which the action is now by isometries. The question is:

If I have some curvature condition on $g$, (specifically, I'm interested in the case of positive sectional curvature), is it possible to preserve the curvature condition after taking the average?

Answers may include include extra hypothesis on $G$, $\mu$, etc.

Let $M$ be a finite dimensional smooth manifold endowed with a riemannian metric $g$ and a smooth action $\mu$ by a compact Lie group $G$. Averaging $g$ over $G$ defines a new metric $$g'(X,Y)=\int_Gg(d\mu_aX,d\mu_aY)da,$$ (integral with respect to a invariant Haar measure) for which the action is now by isometries. The question is:

If I have some curvature condition on $g$, (specifically, I'm interested in the case of positive sectional curvature), is it possible to preserve the curvature condition after taking the average?

Answers may include extra hypothesis on $G$, $\mu$, $M$, etc, as well as changes in the averaging process.