Injectivity radius of Exponential and curvature

Question

Define : Injectivity radius , Exponential

This question is considered in Riemann manifold. I think the Injectivity radius is connect with curvature.

I guess the Injectivity radius can be controlled by curvature.I think there should be some function make the below inequality right. $$ f(curvature)\leq \text{Injectivity radius} \leq g(curvature) $$

Is it right? Or there is other better conclusion ? What I should read about this question ?

Answer 1

There is no such control. For example, for each $r_1, r_2 >0$, the torus

$$\{(r_1e^{i\theta_1}, r_2e^{i\theta_2} ): \theta_1, \theta_2 \in \mathbb R\} \in \mathbb C^2$$

has zero curvature, however by varying $r_1, r_2$, the injectivity radius can be made arbitrarily large/small.

What the curvature really control is the conjugate locus, which is a subset of the cut points.