Submanifolds

Differential Geometry · Axiom Academy

Understanding embedded and immersed submanifolds in differential geometry Embedded submanifolds are "nice" subsets that locally look like coordinate slices. They inherit a natural manifold structure from the ambient space and the inclusion map is a smooth embedding (injective immersion that is a homeomorphism onto its image). The subspace topology coincides with the manifold topology No self-intersections or "bad" behavior Locally looks like a linear subspace Immersed submanifolds allow more flexibility than embedded ones. The key difference is that the topology on M may be finer than the subspace topology inherited from N . Embedded: injective immersion + homeomorphism onto image Immersed: just requires differential to be injective Immersed submanifolds can have self-intersections Classic example: figure-eight curve (immersed but not embedded) This theorem is one of the most powerful tools for constructing submanifolds. It tells us that level sets of smooth functions are submanifolds, provided the differential is surjective. Proving that spheres, ellipsoids, and other level sets are manifolds Understanding constraint manifolds in mechanics Constructing examples in differential topology 4. Example: The Sphere S² ⊂ ℝ³ The unit sphere in ℝ³ is the quintessential example of an embedded submanifold. Let's see why using the Regular Value Theorem. Define f: ℝ³ → ℝ by f(x, y, z) = x² + y² + z² . Then: The differential is df = (2x, 2y, 2z)

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