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Suppose I have a smooth, noncompact manifold $M$ with metrics $g_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that $$ C^{-1} d_1(x,y) \leq d_2(x,y) \leq C d_1(x,y)$$ where $d_i$ are the induced distance metric of $g_i$. Then, if the injectivity radius with respect to $g_1$ satisfies $\mathrm{inj}(M, g_1) \geq \kappa > 0$, does it follow that $\mathrm{inj}(M, g_2) \geq \kappa' > 0$ for some $\kappa'$?

I think I've confused and misled some people by introducing the completely irrelevant $\kappa$ and $\kappa'$. Sorry about that. What I mean is that if $\mathrm{inj}(M,g_1) > 0$ then does it follow that $\mathrm{inj}(M,g_2) > 0$?

Suppose I have a smooth, noncompact manifold $M$ with metrics $g_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that $$ C^{-1} d_1(x,y) \leq d_2(x,y) \leq C d_1(x,y)$$ where $d_i$ are the induced distance metric of $g_i$. Then, if the injectivity radius with respect to $g_1$ satisfies $\mathrm{inj}(M, g_1) \geq \kappa > 0$, does it follow that $\mathrm{inj}(M, g_2) \geq \kappa' > 0$ for some $\kappa'$?

I think I've confused and misled some people by introducing the completely irrelevant $\kappa$ and $\kappa'$. Sorry about that. What I mean is that if $\mathrm{inj}(M,g_1) > 0$ then does it follow that $\mathrm{inj}(M,g_2) > 0$?

Note also that $\mathrm{inj}(M,g)$ is the infimum of the injectivity radius with respect to $g$ at each point.

Suppose I have a smooth, noncompact manifold $M$ with metrics $g_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that $$ C^{-1} d_1(x,y) \leq d_2(x,y) \leq C d_1(x,y)$$ where $d_i$ are the induced distance metric of $g_i$. Then, if the injectivity radius with respect to $g_1$ satisfies $\mathrm{inj}(M, g_1) \geq \kappa > 0$, does it follow that $\mathrm{inj}(M, g_2) \geq \kappa' > 0$ for some $\kappa'$?

The $\kappa$ here doesn't matter. It actually just confuses the question. What I mean is that if $\mathrm{inj}(M,g_1) > 0$ then does it follow that $\mathrm{inj}(M,g_2) > 0$?

Suppose I have a smooth, noncompact manifold $M$ with metrics $g_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that $$ C^{-1} d_1(x,y) \leq d_2(x,y) \leq C d_1(x,y)$$ where $d_i$ are the induced distance metric of $g_i$. Then, if the injectivity radius with respect to $g_1$ satisfies $\mathrm{inj}(M, g_1) \geq \kappa > 0$, does it follow that $\mathrm{inj}(M, g_2) \geq \kappa' > 0$ for some $\kappa'$?

I think I've confused and misled some people by introducing the completely irrelevant $\kappa$ and $\kappa'$. Sorry about that. What I mean is that if $\mathrm{inj}(M,g_1) > 0$ then does it follow that $\mathrm{inj}(M,g_2) > 0$?

Suppose I have a smooth manifold $M$ with metrics $g_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that $$ C^{-1} d_1(x,y) \leq d_2(x,y) \leq C d_1(x,y)$$ where $d_i$ are the induced distance metric of $g_i$. Then, if the injectivity radius with respect to $g_1$ satisfies $\mathrm{inj}(M, g_1) \geq \kappa > 0$, does it follow that $\mathrm{inj}(M, g_2) \geq \kappa' > 0$ for some $\kappa'$?

Suppose I have a smooth, noncompact manifold $M$ with metrics $g_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that $$ C^{-1} d_1(x,y) \leq d_2(x,y) \leq C d_1(x,y)$$ where $d_i$ are the induced distance metric of $g_i$. Then, if the injectivity radius with respect to $g_1$ satisfies $\mathrm{inj}(M, g_1) \geq \kappa > 0$, does it follow that $\mathrm{inj}(M, g_2) \geq \kappa' > 0$ for some $\kappa'$?

The $\kappa$ here doesn't matter. It actually just confuses the question. What I mean is that if $\mathrm{inj}(M,g_1) > 0$ then does it follow that $\mathrm{inj}(M,g_2) > 0$?