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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.

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    $\begingroup$ Curvature bounds tend to get destroyed by averaging. For example, most of the current research in positive sectional curvature is on classifying manifolds with large isometry groups, and there are many Lie group actions which cannot preserve positively/nonnegatively curved metrics. See e.g. the surveys at math.upenn.edu/~wziller/research.html. $\endgroup$
    – Igor Belegradek
    Commented Sep 11, 2016 at 12:01

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As mentioned by Igor Belegradek, the curvature gets destroyed by averaging.

Assume there is a modified averaging process that preserves positive curvature. Then you the Hopf conjecture (there is no positively curved metric on $\mathbb{S}^2\times\mathbb{S}^2$) would follow from the Hsiang--Kleiner result. So at least it is too much to expect.

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