Submersion (mathematics)

 ( Maps Of Manifolds ) 

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential pushforward is everywhere surjective. It is a basic concept in differential topology, dual to that of an immersion.

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Definition Let M and N be differentiable manifolds, and let $f\backslash colon\; M\backslash to\; N$ be a differentiable map between them. The map is a submersion at a point $p\; \backslash in\; M$ if its differential

$Df\_p\; \backslash colon\; T\_p\; M\; \backslash to\; T\_\{f(p)\}N$

is a surjective linear map.. . . . . . . In this case, is called a regular point of the map ; otherwise, is a critical point. A point $q\; \backslash in\; N$ is a regular value of if all points in the preimage $f^\{-1\}(q)$ are regular points. A differentiable map that is a submersion at each point $p\; \backslash in\; M$ is called a submersion. Equivalently, is a submersion if its differential $Df\_p$ has constant rank equal to the dimension of .

Some authors use the term critical point to describe a point where the rank of the Jacobian matrix of at is not maximal.:. Indeed, this is the more useful notion in singularity theory. If the dimension of is greater than or equal to the dimension of , then these two notions of critical point coincide. However, if the dimension of is less than the dimension of , all points are critical according to the definition above (the differential cannot be surjective), but the rank of the Jacobian may still be maximal (if it is equal to dim ). The definition given above is the more commonly used one, e.g., in the formulation of Sard's theorem.

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Submersion theorem Given a submersion $f\backslash colon\; M\backslash to\; N$ between smooth manifolds of dimensions $m$ and $n$, for each $x\; \backslash in\; M$ there exist surjective charts $\backslash phi\; :\; U\; \backslash to\; \backslash mathbb\{R\}^m$ of $M$ around $x$, and $\backslash psi\; :\; V\; \backslash to\; \backslash mathbb\{R\}^n$ of $N$ around $f(x)$, such that $f$ restricts to a submersion $f\; \backslash colon\; U\; \backslash to\; V$ which, when expressed in coordinates as $\backslash psi\; \backslash circ\; f\; \backslash circ\; \backslash phi^\{-1\}\; :\; \backslash mathbb\{R\}^m\; \backslash to\; \backslash mathbb\{R\}^n$, becomes an ordinary orthogonal projection. As an application, for each $p\; \backslash in\; N$ the corresponding fiber of $f$, denoted $M\_p\; =\; f^\{-1\}(\{p\})$ can be equipped with the structure of a smooth submanifold of $M$ whose dimension equals the difference of the dimensions of $N$ and $M$.

This theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure).

For example, consider $f\backslash colon\; \backslash mathbb\{R\}^3\; \backslash to\; \backslash mathbb\{R\}$ given by $f(x,y,z)\; =\; x^4\; +\; y^4\; +z^4.$. The Jacobian matrix is

This has maximal rank at every point except for $(0,0,0)$. Also, the fibers

$f^\{-1\}(\backslash \{t\backslash \})\; =\; \backslash left\backslash \{(a,b,c)\backslash in\; \backslash mathbb\{R\}^3\; :\; a^4\; +\; b^4\; +\; c^4\; =\; t\backslash right\backslash \}$

are empty set for , and equal to a point when $t\; =\; 0$. Hence, we only have a smooth submersion $f\backslash colon\; \backslash mathbb\{R\}^3\backslash setminus\; \{(0,0,0)\}\backslash to\; \backslash mathbb\{R\}\_\{>0\},$ and the subsets $M\_t\; =\; \backslash left\backslash \{(a,b,c)\backslash in\; \backslash mathbb\{R\}^3\; :\; a^4\; +\; b^4\; +\; c^4\; =\; t\backslash right\backslash \}$ are two-dimensional smooth manifolds for $t\; >\; 0$.

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Examples

• Any projection $\backslash pi\backslash colon\; \backslash mathbb\{R\}^\{m+n\}\; \backslash rightarrow\; \backslash mathbb\{R\}^n\backslash subset\; \backslash mathbb\{R\}^\{m+n\}$
• Local diffeomorphisms
• Riemannian submersions
• The projection in a smooth vector bundle or a more general smooth fibration. The surjectivity of the differential is a necessary condition for the existence of a local trivialization.

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Maps between spheres A large class of examples of submersions are submersions between spheres of higher dimension, such as

$f:S^\{n+k\}\; \backslash to\; S^k$

whose fibers have dimension $n$. This is because the fibers (inverse images of elements $p\; \backslash in\; S^k$) are smooth manifolds of dimension $n$. Then, if we take a path

$\backslash gamma:\; I\; \backslash to\; S^k$

and take the pullback

$\backslash begin\{matrix\}$

M_I & \to & S^{n+k} \\ \downarrow & & \downarrow f \\ I & x\rightarrow{\gamma} & S^k \end{matrix} we get an example of a special kind of Cobordism, called a framed bordism. In fact, the framed cobordism groups $\backslash Omega\_n^\{fr\}$ are intimately related to the stable homotopy groups.

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Families of algebraic varieties Another large class of submersions is given by families of algebraic varieties $\backslash pi:\backslash mathfrak\{X\}\; \backslash to\; S$ whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstrass family $\backslash pi:\backslash mathcal\{W\}\; \backslash to\; \backslash mathbb\{A\}^1$ of Elliptic curve is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory, such as intersection homology and Perverse sheaf. This family is given by

$\backslash mathcal\{W\}\; =\; \backslash left\backslash \{(t,x,y)\; \backslash in\; \backslash mathbb\{A\}^1\backslash times\; \backslash mathbb\{A\}^2\; :\; y^2\; =\; x(x-1)(x-t)\; \backslash right\backslash \}$

where $\backslash mathbb\{A\}^1$ is the affine line and $\backslash mathbb\{A\}^2$ is the affine plane. Since we are considering complex varieties, these are equivalently the spaces $\backslash mathbb\{C\},\backslash mathbb\{C\}^2$ of the complex line and the complex plane. Note that we should actually remove the points $t\; =\; 0,1$ because there are singularities (since there is a double root).

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Local normal form If is a submersion at and , then there exists an open neighborhood of in , an open neighborhood of in , and local coordinates at and at such that , and the map in these local coordinates is the standard projection

$f(x\_1,\; \backslash ldots,\; x\_n,\; x\_\{n+1\},\; \backslash ldots,\; x\_m)\; =\; (x\_1,\; \backslash ldots,\; x\_n).$

It follows that the full preimage in of a regular value in under a differentiable map is either empty or a differentiable manifold of dimension , possibly connected space. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all in if the map is a submersion.

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Topological manifold submersions Submersions are also well-defined for general topological manifolds.. A topological manifold submersion is a continuous surjection such that for all in , for some continuous charts at and at , the map is equal to the projection map from to , where .

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See also

• Ehresmann's fibration theorem

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Notes

• (1985). 9780817631871, Birkhäuser. ISBN 9780817631871
• James W. Bruce (1984). 9780521429993, Cambridge University Press. ISBN 9780521429993
• (1994). 9780521231909, Cambridge University Press. . ISBN 9780521231909
• Manfredo Perdigao do Carmo (1994). 9780817634902 ISBN 9780817634902
• Theodore Frankel (1997). 9780521387538, Cambridge University Press. ISBN 9780521387538
• (2026). 9783540204930, Springer-Verlag. ISBN 9783540204930
• Antoni Albert Kosinski (2026). 9780486462448, Dover Publications. ISBN 9780486462448
• Serge Lang (1999). 9780387985930, Springer. ISBN 9780387985930
• (2026). 9780486478555, Dover Publications. ISBN 9780486478555

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Further reading

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