Suppose M is a compact manifold and Rim(M) the space of all Riemannian metrics. Consider $L^2$ metric $ G_g(h,k) = \int_M tr(g^{-1}hg^{-1}k)vol(g)$ where is a Riemannian metric and $ h,k \in T_gM$ . What is known about the injectivity radius of this metric? for example is it uniformly positive?
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$\begingroup$ What is $g$? Since you write $T_g M$, it seems to be a point of $M$; why does it have an inverse, is $M$ a Lie group? What does that trace mean? Where does one use $Rim (M)$? What is $vol(g)$? $\endgroup$– Alex M.Commented Feb 25, 2016 at 12:13
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$\begingroup$ By $T_gM$ you mean $T_gRiem(M)$ ? So you're asking about the injectivity radius of an infinite dimensional Riemannian manifold ? I suspect the situation can be pretty bad. $\endgroup$– Thomas RichardCommented Feb 25, 2016 at 14:02
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$\begingroup$ Sorry, I misspelled . g is a metric and $ T_g$ must be replaced by $T_g Riem (M)$. It is infinite dimensional Frechet manifold. The only thing that I found about injectivity radius is that it is discontinuous. $\endgroup$– KavehCommented Feb 29, 2016 at 9:01
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