“Preimage theorem”的意思、由来-开放百科全书

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Preimage theorem

释义

Statement of Theorem
References

In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map.^[1]^[2]

Statement of Theorem

Definition. Let be a smooth map between manifolds. We say that a point is a regular value of f if for all the map is surjective. Here, and are the tangent spaces of X and Y at the points x and y.

Theorem. Let be a smooth map, and let be a regular value of f; then is a submanifold of X. If , then the codimension of is equal to the dimension of Y. Also, the tangent space of at is equal to .

References

1. ^{{citation|title=An Introduction to Manifolds|first=Loring W.|last=Tu|publisher=Springer|year=2010|isbn=9781441974006|contribution=9.3 The Regular Level Set Theorem|pages=105–106|url=https://books.google.com/books?id=xQsTJJGsgs4C&pg=PA105}}.
2. ^{{citation|title=Lectures on Morse Homology|volume=29|series=Texts in the Mathematical Sciences|first=Augustin|last=Banyaga|publisher=Springer|year=2004|isbn=9781402026959|page=130|url=https://books.google.com/books?id=AX-_sbMjOK4C&pg=PA130|contribution=Corollary 5.9 (The Preimage Theorem)}}.

1 : Theorems in differential topology

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