Submersion is a smooth and surjective differential map between differentiable manifolds, where the differential at each point is surjective. This concept is vital in understanding how one manifold can be 'mapped down' onto another, preserving certain geometric structures. Submersions are particularly important in the context of studying Riemannian submersions and the behavior of embedded and immersed submanifolds, as they provide insights into how different geometric properties interact when transitioning from one manifold to another.
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A submersion allows for local sections, meaning around any point, there is a neighborhood that looks like a product of the base manifold and a Euclidean space.
The rank of the differential of a submersion is equal to the dimension of the target manifold, ensuring that the image can cover open sets of the target space.
Submersions help define the notion of fiber bundles, where fibers over points in the base manifold can be viewed as the preimages under the submersion map.
In Riemannian geometry, submersions help analyze geodesics and curvature by providing a way to project metrics from one manifold onto another.
Submersions can be classified into regular and singular types depending on whether their fibers are compact or not, impacting their geometric properties.
Review Questions
How does the concept of submersion relate to the construction of fiber bundles in differential geometry?
Submersion plays a crucial role in constructing fiber bundles by defining how one manifold can be mapped onto another while maintaining certain structural properties. When a manifold is submerged onto another, each point in the base space corresponds to a fiber that consists of points in the total space. This correspondence illustrates how fibers can be understood through the lens of submersions, allowing for a better grasp of both local and global geometric structures within fiber bundles.
Discuss how Riemannian submersions utilize the concept of submersion to preserve geometric properties when mapping between manifolds.
Riemannian submersions extend the idea of submersion by ensuring that not only is the map smooth and surjective, but it also preserves distances and angles. This means that geodesics on the total space project down to geodesics on the base manifold, maintaining important metric properties. This preservation is crucial for applications in physics and other fields where understanding curved spaces is essential, showcasing how geometry can be interconnected through this mapping process.
Evaluate the implications of singular vs. regular submersions on the study of immersed submanifolds and their geometric structures.
The distinction between regular and singular submersions significantly affects the study of immersed submanifolds, particularly regarding their geometric characteristics. Regular submersions have well-behaved fibers that are typically compact, leading to more straightforward topological features and analysis. In contrast, singular submersions may present challenges due to their non-compact fibers or critical points where typical properties break down. Understanding these implications allows mathematicians to better navigate complex geometries while exploring phenomena such as curvature and stability in immersed structures.
Related terms
Riemannian Submersion: A specific type of submersion where a Riemannian manifold is projected onto another Riemannian manifold in a way that preserves lengths of curves and angles between them.
Immersed Submanifold: A subset of a manifold that is locally diffeomorphic to Euclidean space but may not be globally equivalent, often arising in discussions of submersions.
A function between manifolds that is continuously differentiable, meaning it has derivatives of all orders and maintains differentiability properties during mapping.