Let $x\in M$ be a point, whose injectivity radius is $r_x$. So is it true that for any point $y \in B(x, r_x)$, the injectivity radius at $y$ is at least $r_x- d(x, y)$? is there any book has this result or if it is false what is the count example?
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@tessellation: I'm not seeing how that's a counterexample. – Anthony Carapetis Nov 14 '13 at 10:16
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@anthony oops! I am sorry its a mistake. – tessellation Nov 14 '13 at 10:21
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This is not true. Consider a smooth surface of revolution that looks something like a long cylinder of radius $r$ and length $L$ capped off by hemispheres. (If you care about the fact that this surface is only $C^1$, you can smooth it out to $C^\infty$ while keeping it a surface of revolution and the argument will hold).
Let $x$ be the point up top. Then it's easy enough to see from symmetry that the injectivity radius at $x$ is $L + \pi r$; i.e. the exponential map can be restricted to a disc such that it is a diffeomorphism onto all of the surface except the antipodal (bottom) point. In contrast, for any point $y$ on the cylindrical portion of the surface the injectivity radius is just $\pi r$, since the circles of revolution are geodesic loops. Choose $L \gg r$ and $y$ quite close to $x$ on the cylindrical portion and you have a counterexample.
Anthony Carapetis
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I see, thanks. btw, what is the software that you used to draw this beautifule picture? – Sun Nov 14 '13 at 12:29
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