Submersion (mathematics)

{{Short description|Differential map between manifolds whose differential is everywhere surjective}}{{redirect|Regular point|"regular point of an algebraic variety"|Singular point of an algebraic variety}} 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.

== Definition == Let ''M'' and ''N'' be differentiable manifolds, and let <math>f\colon M\to N</math> be a differentiable map between them. The map {{math|''f''}} is a '''submersion at a point''' <math>p \in M</math> if its differential

:<math>Df_p \colon T_p M \to T_{f(p)}N</math>

is a surjective linear map.<ref>{{harvnb|Crampin|Pirani|1994|page=243}}. {{harvnb|do Carmo|1994|page=185}}. {{harvnb|Frankel|1997|page=181}}. {{harvnb|Gallot|Hulin|Lafontaine|2004|page=12}}. {{harvnb|Kosinski|2007|page=27}}. {{harvnb|Lang|1999|page=27}}. {{harvnb|Sternberg|2012|page=378}}.</ref> In this case, {{math|''p''}} is called a '''regular point''' of the map {{math|''f''}}; otherwise, {{math|''p''}} is a ''critical point''. A point <math>q \in N</math> is a '''regular value''' of {{math|''f''}} if all points {{math|''p''}} in the preimage <math>f^{-1}(q)</math> are regular points. A differentiable map {{math|''f''}} that is a submersion at each point <math>p \in M</math> is called a '''submersion'''. Equivalently, {{math|''f''}} is a submersion if its differential <math>Df_p</math> has constant rank equal to the dimension of {{math|''N''}}.

Some authors use the term ''critical point'' to describe a point where the rank of the Jacobian matrix of {{math|''f''}} at {{math|''p''}} is not maximal.:<ref>{{harvnb|Arnold|Gusein-Zade|Varchenko|1985}}.</ref> Indeed, this is the more useful notion in singularity theory. If the dimension of {{math|''M''}} is greater than or equal to the dimension of {{math|''N''}}, then these two notions of critical point coincide. However, if the dimension of {{math|''M''}} is less than the dimension of {{math|''N''}}, 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 {{math|''M''}}). The definition given above is the more commonly used one, e.g., in the formulation of Sard's theorem.

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

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

For example, consider <math>f\colon \mathbb{R}^3 \to \mathbb{R}</math> given by <math>f(x,y,z) = x^4 + y^4 +z^4.</math>. The Jacobian matrix is :<math>\begin{bmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z} \end{bmatrix} = \begin{bmatrix} 4x^3 & 4y^3 & 4z^3 \end{bmatrix}.</math>

This has maximal rank at every point except for <math>(0,0,0)</math>. Also, the fibers :<math>f^{-1}(\{t\}) = \left\{(a,b,c)\in \mathbb{R}^3 : a^4 + b^4 + c^4 = t\right\}</math>

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

== Examples == * Any projection <math>\pi\colon \mathbb{R}^{m+n} \rightarrow \mathbb{R}^n\subset \mathbb{R}^{m+n}</math> * 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.

=== Maps between spheres === A large class of examples of submersions are submersions between spheres of higher dimension, such as :<math>f:S^{n+k} \to S^k</math> whose fibers have dimension <math>n</math>. This is because the fibers (inverse images of elements <math>p \in S^k</math>) are smooth manifolds of dimension <math>n</math>. Then, if we take a path :<math>\gamma: I \to S^k</math> and take the pullback :<math>\begin{matrix} M_I & \to & S^{n+k} \\ \downarrow & & \downarrow f \\ I & x\rightarrow{\gamma} & S^k \end{matrix}</math> we get an example of a special kind of bordism, called a framed bordism. In fact, the framed cobordism groups <math>\Omega_n^{fr}</math> are intimately related to the stable homotopy groups.

=== Families of algebraic varieties === Another large class of submersions is given by families of algebraic varieties <math>\pi:\mathfrak{X} \to S</math> whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstrass family <math>\pi:\mathcal{W} \to \mathbb{A}^1</math> of elliptic curves is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory, such as intersection homology and perverse sheaves. This family is given by<blockquote><math>\mathcal{W} = \left\{(t,x,y) \in \mathbb{A}^1\times \mathbb{A}^2 : y^2 = x(x-1)(x-t) \right\}</math></blockquote>where <math>\mathbb{A}^1</math> is the affine line and <math>\mathbb{A}^2</math> is the affine plane. Since we are considering complex varieties, these are equivalently the spaces <math>\mathbb{C},\mathbb{C}^2</math> of the complex line and the complex plane. Note that we should actually remove the points <math>t = 0,1</math> because there are singularities (since there is a double root).

== Local normal form == If {{math|''f'': ''M'' → ''N''}} is a submersion at {{math|''p''}} and {{math|''f''(''p'') {{=}} ''q'' ∈ ''N''}}, then there exists an open neighborhood {{math|''U''}} of {{math|''p''}} in {{math|''M''}}, an open neighborhood {{math|''V''}} of {{math|''q''}} in {{math|''N''}}, and local coordinates {{math|(''x''1, …, ''x''''m'')}} at {{math|''p''}} and {{math|(''x''1, …, ''x''''n'')}} at {{math|''q''}} such that {{math|''f''(''U'') {{=}} ''V''}}, and the map {{math|''f''}} in these local coordinates is the standard projection : <math>f(x_1, \ldots, x_n, x_{n+1}, \ldots, x_m) = (x_1, \ldots, x_n).</math>

It follows that the full preimage {{math|''f''−1(''q'')}} in {{math|''M''}} of a regular value {{math|''q''}} in {{math|''N''}} under a differentiable map {{math|''f'': ''M'' → ''N''}} is either empty or a differentiable manifold of dimension {{math|dim ''M'' − dim ''N''}}, possibly disconnected. This is the content of the '''regular value theorem''' (also known as the '''submersion theorem'''). In particular, the conclusion holds for all {{math|''q''}} in {{math|''N''}} if the map {{math|''f''}} is a submersion.

== Topological manifold submersions == Submersions are also well-defined for general topological manifolds.<ref>{{harvnb|Lang|1999|page=27}}.</ref> A topological manifold submersion is a continuous surjection {{math|''f'' : ''M'' → ''N''}} such that for all {{math|''p''}} in {{math|''M''}}, for some continuous charts {{math|ψ}} at {{math|''p''}} and {{math|φ}} at {{math|''f(p)''}}, the map {{math|''ψ−1 ∘ f ∘ φ''}} is equal to the projection map from {{math|'''R'''''m''}} to {{math| '''R'''n}}, where {{math|''m'' {{=}} dim(''M'') ≥ ''n'' {{=}} dim(''N'')}}.

== See also == * Ehresmann's fibration theorem

==Notes== {{reflist}}

== References ==

* {{cite book|first1=Vladimir I.|last1=Arnold|author-link1=Vladimir Arnold|first2=Sabir M.|last2=Gusein-Zade|author-link2=Sabir Gusein-Zade|first3=Alexander N.|last3=Varchenko|author-link3=Alexander Varchenko|title=Singularities of Differentiable Maps: Volume 1|publisher=Birkhäuser|year=1985|ISBN=0-8176-3187-9}} * {{cite book|first=James W.|last=Bruce|first2=Peter J.|last2=Giblin|title=Curves and Singularities|publisher=Cambridge University Press|year=1984|ISBN=0-521-42999-4|mr=0774048}} * {{cite book|last1=Crampin|first1=Michael|last2=Pirani|first2=Felix Arnold Edward|title=Applicable differential geometry|publisher=Cambridge University Press|location=Cambridge, England|year=1994|isbn=978-0-521-23190-9|url-access=registration|url=https://archive.org/details/applicablediffer0000cram}} * {{cite book|title = Riemannian Geometry|first=Manfredo Perdigao | last = do Carmo |author-link=Manfredo do Carmo | year = 1994|isbn=978-0-8176-3490-2}} * {{cite book|last=Frankel|first=Theodore|title=The Geometry of Physics|publisher=Cambridge University Press|location=Cambridge|year=1997|isbn=0-521-38753-1|mr=1481707}} * {{cite book| last1=Gallot | first1=Sylvestre | last2=Hulin | first2=Dominique|author2-link=Dominique Hulin | last3=Lafontaine | first3=Jacques | title=Riemannian Geometry | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-3-540-20493-0 | year=2004}} * {{cite book|last=Kosinski|first=Antoni Albert|year=2007|orig-year=1993|title=Differential manifolds|location=Mineola, New York|publisher=Dover Publications|isbn=978-0-486-46244-8}} * {{cite book| isbn = 978-0-387-98593-0 | title = Fundamentals of Differential Geometry | last = Lang | first = Serge |author-link=Serge Lang|publisher=Springer|location=New York| year = 1999 | series = Graduate Texts in Mathematics}} * {{cite book|last1=Sternberg|first1=Shlomo Zvi|author-link1=Shlomo Sternberg|year=2012|title=Curvature in Mathematics and Physics|publisher=Dover Publications|location=Mineola, New York|isbn=978-0-486-47855-5}}

== Further reading == *https://mathoverflow.net/questions/376129/what-are-the-sufficient-and-necessary-conditions-for-surjective-submersions-to-b?rq=1

Category:Maps of manifolds Category:Smooth functions