Regular Point Theorem of Manifolds with Boundaries

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The regular value theorem in differential geometry traditionally applies to manifolds without boundaries, stating that the preimage of a regular value is an imbedding submanifold. The discussion raises the question of how this theorem extends to manifolds with boundaries. Milnor's "Topology from the Differentiable Viewpoint" is recommended as a resource for understanding this topic. Additionally, the regular value theorem is linked to the Implicit Function Theorem, suggesting that extending it to manifolds with boundaries is a valuable exercise. This highlights the need for further exploration of the theorem's implications in the context of boundaries.
Fangyang Tian
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Dear Folks:
In most textbooks on differential geometry, the regular theorem states for manifolds without boundaries: the preimage of a regular value is a imbedding submanifold. What about the monifolds with boundaries??
Many Thanks!
 
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Fangyang Tian said:
Dear Folks:
In most textbooks on differential geometry, the regular theorem states for manifolds without boundaries: the preimage of a regular value is a imbedding submanifold. What about the monifolds with boundaries??
Many Thanks!

I passionately recommend Milnor's book, Topology from the Differentiable Viewpoint which covers this and many other topics in Differential Topology.

The regular value theorem is an application of the Implicit Function Theorem. Extending it to a manifold with boundary is a good exercise.
 

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