Injectivity Radius: Continuity On Incomplete Manifolds

Aug 25, 2025 by Lucas 55 views

Exploring Injectivity Radius Continuity on Incomplete Riemannian Manifolds

Hey everyone! Let's dive into a fascinating area of Riemannian geometry: the injectivity radius on non-complete Riemannian manifolds. Specifically, we're going to discuss whether the injectivity radius function is (semi) continuous in this context. This is quite an exciting topic, and hopefully, we can unravel some of its mysteries together. Let's get started!

Defining the Injectivity Radius

Before we get into the nitty-gritty details, let's define our terms clearly. The injectivity radius, denoted as inj(p) for a point p on a Riemannian manifold M, is essentially the largest radius r such that the exponential map exp_p is a diffeomorphism from the open ball of radius r in the tangent space T_pM to its image in M. In simpler terms, it tells us how far we can "shoot out" geodesics from a point p before they start intersecting or behaving strangely.

Formally, the injectivity radius at a point p in M is defined as:

inj(p) = sup {r > 0 : exp_p|_{B_r(0)} : B_r(0) \subset T_pM -> M is a diffeomorphism onto its image`}

Where B_r(0) is the open ball of radius r centered at the origin in the tangent space T_pM. If M is complete, then inj(p) is also the distance to the closest cut point of p. The cut locus of p is the set of points q on M such that there exists a geodesic from p to q that minimizes distance, but this geodesic is not unique, or it is not minimizing beyond q.

Understanding the injectivity radius is crucial because it gives us a measure of how "well-behaved" the local geometry around a point is. A larger injectivity radius implies a more predictable geodesic behavior, which is often desirable in many geometric and topological arguments. It's like having a clear map of the neighborhood around your house; the larger the area you know well, the easier it is to navigate. When the injectivity radius is small, geodesics can start looping back on themselves or intersecting in complicated ways, making the geometry much harder to analyze. This concept is used extensively in various areas, including the study of geodesic flows, curvature bounds, and the topology of manifolds. So, next time you think about the injectivity radius, remember it as a gauge of geometric predictability and stability. It's one of those fundamental concepts that quietly underpins a lot of deeper results in Riemannian geometry.

Continuity on Complete Riemannian Manifolds

Now, here’s a well-known fact: if our Riemannian manifold M is complete and connected, then the injectivity radius function inj : M -> (0, ∞] is continuous. Completeness here means that all geodesics can be infinitely extended. This is a significant property because it allows us to make certain analytical arguments about the injectivity radius. For example, knowing that the injectivity radius varies continuously lets us find minimum values on compact sets and make conclusions about the global geometry of the manifold.

However, the question becomes much more interesting when we relax the completeness assumption. What happens if M is not complete? Can we still say something about the (semi) continuity of the injectivity radius function? This is where things get a bit more challenging and requires a more nuanced approach. Think of it like this: completeness provides a sort of 'safety net' that ensures nice behavior of geodesics, and when we remove that net, things can get a bit wild. For example, in a non-complete manifold, geodesics might reach the boundary in finite time, leading to sudden drops in the injectivity radius. The absence of completeness can lead to behaviors that are not seen in complete manifolds, such as the injectivity radius being discontinuous at certain points. Understanding these behaviors is crucial for analyzing the local and global geometry of non-complete Riemannian manifolds, which often arise in various contexts, including the study of manifolds with singularities or boundaries.

The Challenge of Incompleteness

When M is not complete, the injectivity radius function can indeed lose its continuity. This is because geodesics might hit the "edge" of the manifold, causing the injectivity radius to drop abruptly. Understanding the behavior of the injectivity radius in such cases requires a deeper dive into the properties of non-complete manifolds.

Examples of Discontinuity

To illustrate this, consider some examples. Imagine a disk with a point removed. Near that removed point, the injectivity radius will tend to zero, but it might not do so continuously. Or, think of a surface with a cusp-like singularity. As you approach the cusp, the injectivity radius can behave erratically. These examples highlight that incompleteness introduces complexities that are absent in complete manifolds. It's like trying to predict the stock market; when everything is stable, you can make reasonable forecasts, but when unexpected events happen (like incompleteness), things become much harder to predict.

Semi-Continuity

Even if the injectivity radius is not continuous, we might still ask if it is semi-continuous. Recall that a function f is lower semi-continuous if liminf x->a f(x) >= f(a) and upper semi-continuous if limsup x->a f(x) <= f(a). Determining whether the injectivity radius satisfies either of these conditions can provide some valuable information about its behavior, even in the absence of full continuity.

Exploring Further

So, is the injectivity radius (semi) continuous on a non-complete Riemannian manifold? The answer isn't straightforward and depends heavily on the specific manifold in question. While completeness guarantees continuity, incompleteness can lead to discontinuities. However, exploring semi-continuity might offer some insights.

Here are some avenues for further investigation:

Specific Examples: Examine specific non-complete manifolds and try to explicitly compute the injectivity radius function. Look for examples where the injectivity radius is discontinuous or semi-continuous but not continuous.
Conditions for Semi-Continuity: Are there additional conditions we can impose on the non-complete manifold to ensure that the injectivity radius is at least semi-continuous? For instance, what if the manifold is "almost complete" in some sense?
Relationship to Curvature: How does the curvature of the manifold affect the continuity or semi-continuity of the injectivity radius? Are there curvature bounds that can guarantee some form of continuity?

By exploring these questions, we can gain a deeper understanding of the injectivity radius and its behavior on non-complete Riemannian manifolds. It's a fascinating area with plenty of open questions, and I encourage you to delve into it further!

Conclusion

In summary, while the injectivity radius is continuous on complete Riemannian manifolds, this property doesn't necessarily hold for non-complete manifolds. The absence of completeness can lead to discontinuities, making the analysis more challenging. However, exploring semi-continuity and examining specific examples can provide valuable insights into the behavior of the injectivity radius in these cases. It's like being a detective; sometimes, the absence of evidence is itself evidence. Keep exploring, keep questioning, and happy geometerizing, everyone!