Clarifying a result of Klingenberg

Question

I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each point. I am aware of the following result in the literature (Petersen, Riemannian Geometry):

Lemma 6.4.7 (Klingenberg). Let $(M,g)$ be a compact Riemannian manifold with $\mathrm{sec} \leq K$ where $K > 0$. Then $\mathrm{inj}(p) \geq \min \{ \pi/\sqrt{K}, (1/2)(\text{length of shortest geodesic loop based at $p$})\}$

Note that compactness renders the notion of a shortest geodesic loop based at $p$ to be sensible. However, I have also seen the following stated (A question on closed geodesics):

Now what is true (in a complete Riemannian manifold) is that the injectivity radius is the minimum of two numbers:

1. The half of the length of a shortest geodesic loop at $p$,
2. The distance from $p$ to nearest conjugate point at $p$. (This number is called the conjugate radius at $p$ and it is bounded below by $\pi/\sqrt{k}$ where $k$ is an upper section curvature bound of the ambient manifold).

Similarly, in "Finite Propogation Speed, Kernel Estimates for Functions of the Laplace Operator, and the Geometry of Complete Riemannian Manifolds" (Cheeger-Gromov-Taylor), there is the following paragraph:

Let $M^n$ be a complete Riemannian manifold, and $p\in M^n$. Let $\gamma$ be a geodesic loop at $p$ of shortest length $L[\gamma] = 2l$. Let $K_M \leq K$ on $B_l(p)$. Then by a result of Klingenberg (see e.g., [7]), the injectivity radius of the exponential map at $p$ is bounded below by $\min(l, \pi/\sqrt{K})$; if $K\leq 0$ we interpret $\pi/\sqrt{K}$ as $\infty$.

Both of these sources offer references, wherein the manifold is assumed to be compact or the sectional curvature is bounded from below by a positive number. I do not have these properties for my manifold, but I do have sectional curvature bounded from above. Is there a reliable reference for these statements, or is it easy to see why the compact case implies these results? In these statements, the existence of a shortest geodesic loop is ambiguous without compactness; is the wording implying that if there is no shortest geodesic loop then we are in the $\pi/\sqrt{K}$ case, or should "length of a shortest geodesic loop" really refer to an infimum of lengths of geodesic loops?

Answer 1

If your manifold is complete, there is no problem at all. One way to see this is to analyze the techniques used to see that one only uses local properties to construct the exponential map, study the Jacobi fields, etc. Another way to see it is to take a large enough ball (of radius say 10 times the expected value of the injectivity radius) centered at the point $p$ you are interested in, observe that for a generic value of the distance function the distance sphere will be smooth, and then close it up to a compact manifold. This will have the same injectivity radius at $p$ as the original complete one.