Open Submanifolds & Completeness: Proving P=M

Hey everyone! Today, we're diving deep into a fascinating concept in semi-Riemannian geometry. We're going to explore a theorem that connects the completeness of an open submanifold within a connected manifold. Specifically, we'll be dissecting the statement: If an open submanifold P within a manifold M is complete, and M is connected, then P is equal to M. This is problem 7 from chapter 3 of Barrett O'Neill's renowned book on semi-Riemannian geometry. We'll not only break down the theorem but also discuss a potential proof strategy, aiming to make this concept crystal clear. So, buckle up, and let's get started!

Understanding the Key Concepts

Before we jump into the proof, let's make sure we're all on the same page with the core concepts involved. This will help us build a solid foundation for understanding the theorem and its implications.

Manifolds: The Stage for Our Geometry

First off, we have manifolds. Think of a manifold as a space that locally looks like Euclidean space. This means that if you zoom in close enough to any point on the manifold, it will resemble a familiar flat space like a plane or three-dimensional space. However, globally, a manifold can have a much more complex structure, like the surface of a sphere or a torus (donut shape).

Manifolds are the fundamental stage on which our geometric drama unfolds. They provide the setting for curves, surfaces, and other geometric objects to exist. To rigorously define a manifold, we say that a topological manifold is a topological space that is locally Euclidean and Hausdorff, and has a countable base. This means:

• Locally Euclidean: Every point in the space has a neighborhood that is homeomorphic to an open subset of Euclidean space (ℝⁿ). This is the key property that allows us to use the familiar tools of calculus and linear algebra on manifolds.
• Hausdorff: For any two distinct points, there exist disjoint open sets containing them. This ensures that points are well-separated and that the topology is not too "pathological."
• Countable base: The topology has a countable base, meaning there is a countable collection of open sets such that any open set can be written as a union of sets from this collection. This condition ensures that the manifold is not "too large" in a topological sense.

To make a topological manifold into a smooth manifold, we add the requirement of a smooth structure. This means that we can define smooth functions on the manifold, which allows us to do calculus. A smooth structure is defined by an atlas of charts, where each chart is a homeomorphism from an open subset of the manifold to an open subset of Euclidean space, and the transition maps between charts are smooth functions. The smoothness of the transition maps ensures that the notion of smoothness is well-defined on the entire manifold.

Submanifolds: Worlds Within Worlds

Next up are submanifolds. A submanifold is essentially a manifold that's embedded within another manifold. Imagine a curve drawn on the surface of a sphere; that curve is a submanifold of the sphere. More formally, a submanifold P of a manifold M is a subset of M that is itself a manifold, and the inclusion map from P to M is a smooth embedding. This means that P has its own manifold structure, and it sits