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In [[mathematics]], particularly in [[differential topology]], the '''preimage theorem''' is a theorem concerning the preimage of particular points in a [[manifold]] under the action of a [[smooth map]]. | |||
==Statement of Theorem== | |||
''Definition.'' Let <math>f: X \to Y\,\!</math> be a smooth map between manifolds. We say that a point <math>y \in Y</math> is a ''regular value of f'' if for all <math>x \in f^{-1}(y)</math> the map <math>df_x: T_xX \to T_yY\,\!</math> is [[surjective map|surjective]]. Here, <math>T_xX\,\!</math> and <math>T_yY\,\!</math> are the [[tangent space]]s of X and Y at the points x and y. | |||
''Theorem.'' Let <math>f: X \to Y\,\!</math> be a smooth map, and let <math>y \in Y</math> be a regular value of ''f''. Then <math>f^{-1}(y) = \{x \in X : f(x) =y \}</math> is a submanifold of X. Further, if <math>y</math> is in the image of ''f'', the [[codimension]] of this manifold in X is equal to the dimension of Y, and the [[tangent space]] of <math>f^{-1}(y)</math> at a point <math>x</math> is <math>Ker(df_x)</math>. | |||
{{topology-stub}} | |||
[[Category:Theorems in differential topology]] | |||
Revision as of 18:50, 16 December 2013
In mathematics, particularly in differential topology, the preimage theorem is a theorem concerning the preimage of particular points in a manifold under the action of a smooth map.
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. Further, if is in the image of f, the codimension of this manifold in X is equal to the dimension of Y, and the tangent space of at a point is .
