Rank of a smooth map is lower semicontinuous?












0












$begingroup$


Let $F:Mrightarrow N$ be a smooth map between manifolds, $pin M$. Prove that if $operatorname{rank}_{p}F=r$, then there exists a neighborhood $U$ of $p$, such that for $forall qin U$, $operatorname{rank}_{q}Fgeqslant r$.



Could anyone help me? Thanks in advance.






differential-geometry manifolds smooth-manifolds







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    This is actually a linear-algebra fact. The map $mathrm{rank}:M_{mtimes n}(mathbb{R})tomathbb{Z}$ is lower semicontinuous.
    $endgroup$
    – Amitai Yuval
    Dec 19 '18 at 7:32
















0












$begingroup$


Let $F:Mrightarrow N$ be a smooth map between manifolds, $pin M$. Prove that if $operatorname{rank}_{p}F=r$, then there exists a neighborhood $U$ of $p$, such that for $forall qin U$, $operatorname{rank}_{q}Fgeqslant r$.



Could anyone help me? Thanks in advance.






differential-geometry manifolds smooth-manifolds







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    This is actually a linear-algebra fact. The map $mathrm{rank}:M_{mtimes n}(mathbb{R})tomathbb{Z}$ is lower semicontinuous.
    $endgroup$
    – Amitai Yuval
    Dec 19 '18 at 7:32














0












0








0





$begingroup$


Let $F:Mrightarrow N$ be a smooth map between manifolds, $pin M$. Prove that if $operatorname{rank}_{p}F=r$, then there exists a neighborhood $U$ of $p$, such that for $forall qin U$, $operatorname{rank}_{q}Fgeqslant r$.



Could anyone help me? Thanks in advance.






differential-geometry manifolds smooth-manifolds







share|cite|improve this question









$endgroup$




Let $F:Mrightarrow N$ be a smooth map between manifolds, $pin M$. Prove that if $operatorname{rank}_{p}F=r$, then there exists a neighborhood $U$ of $p$, such that for $forall qin U$, $operatorname{rank}_{q}Fgeqslant r$.



Could anyone help me? Thanks in advance.






differential-geometry manifolds smooth-manifolds




differential-geometry manifolds smooth-manifolds






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 19 '18 at 6:51









user450201user450201

718




718








  • 3




    $begingroup$
    This is actually a linear-algebra fact. The map $mathrm{rank}:M_{mtimes n}(mathbb{R})tomathbb{Z}$ is lower semicontinuous.
    $endgroup$
    – Amitai Yuval
    Dec 19 '18 at 7:32














  • 3




    $begingroup$
    This is actually a linear-algebra fact. The map $mathrm{rank}:M_{mtimes n}(mathbb{R})tomathbb{Z}$ is lower semicontinuous.
    $endgroup$
    – Amitai Yuval
    Dec 19 '18 at 7:32








3




3




$begingroup$
This is actually a linear-algebra fact. The map $mathrm{rank}:M_{mtimes n}(mathbb{R})tomathbb{Z}$ is lower semicontinuous.
$endgroup$
– Amitai Yuval
Dec 19 '18 at 7:32




$begingroup$
This is actually a linear-algebra fact. The map $mathrm{rank}:M_{mtimes n}(mathbb{R})tomathbb{Z}$ is lower semicontinuous.
$endgroup$
– Amitai Yuval
Dec 19 '18 at 7:32










1 Answer
1






active

oldest

votes


















1












$begingroup$

Let us choose coordinate charts $(x_{1},ldots,x_{n})$ around $p$ and $(y_{1},ldots,y_{m})$ around $F(p)$. Writing $F=big(F_{1}(x_{1},ldots,x_{n}),ldots,F_{m}(x_{1},ldots,x_{n})big)$, we can express the derivative $d_{p}F$ as a matrix
$$
begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(p) & cdots & frac{partial F_{1}}{partial x_{n}}(p)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(p) & cdots & frac{partial F_{m}}{partial x_{n}}(p)
end{pmatrix}.
$$

By assumption, this matrix has rank $r$, so there is a nonzero minor of size $rtimes r$:
$$
begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(p)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(p)
end{vmatrix}neq 0.
$$
The map
$$
G:qmapsto begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(q)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(q)
end{vmatrix},
$$
is continuous and nonzero at $p$, so there exists a neighborhood $U$ of $p$ such that $G(q)neq 0$ for all $qin U$. This implies that for all $qin U$
$$
d_{q}F=begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(q) & cdots & frac{partial F_{1}}{partial x_{n}}(q)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(q) & cdots & frac{partial F_{m}}{partial x_{n}}(q)
end{pmatrix}
$$
has rank at least $r$, since it has a nonzero $rtimes r$ minor.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Very clear. Really appreciate:)
    $endgroup$
    – user450201
    Dec 20 '18 at 11:11











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3046105%2frank-of-a-smooth-map-is-lower-semicontinuous%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

Let us choose coordinate charts $(x_{1},ldots,x_{n})$ around $p$ and $(y_{1},ldots,y_{m})$ around $F(p)$. Writing $F=big(F_{1}(x_{1},ldots,x_{n}),ldots,F_{m}(x_{1},ldots,x_{n})big)$, we can express the derivative $d_{p}F$ as a matrix
$$
begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(p) & cdots & frac{partial F_{1}}{partial x_{n}}(p)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(p) & cdots & frac{partial F_{m}}{partial x_{n}}(p)
end{pmatrix}.
$$

By assumption, this matrix has rank $r$, so there is a nonzero minor of size $rtimes r$:
$$
begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(p)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(p)
end{vmatrix}neq 0.
$$
The map
$$
G:qmapsto begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(q)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(q)
end{vmatrix},
$$
is continuous and nonzero at $p$, so there exists a neighborhood $U$ of $p$ such that $G(q)neq 0$ for all $qin U$. This implies that for all $qin U$
$$
d_{q}F=begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(q) & cdots & frac{partial F_{1}}{partial x_{n}}(q)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(q) & cdots & frac{partial F_{m}}{partial x_{n}}(q)
end{pmatrix}
$$
has rank at least $r$, since it has a nonzero $rtimes r$ minor.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Very clear. Really appreciate:)
    $endgroup$
    – user450201
    Dec 20 '18 at 11:11
















1












$begingroup$

Let us choose coordinate charts $(x_{1},ldots,x_{n})$ around $p$ and $(y_{1},ldots,y_{m})$ around $F(p)$. Writing $F=big(F_{1}(x_{1},ldots,x_{n}),ldots,F_{m}(x_{1},ldots,x_{n})big)$, we can express the derivative $d_{p}F$ as a matrix
$$
begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(p) & cdots & frac{partial F_{1}}{partial x_{n}}(p)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(p) & cdots & frac{partial F_{m}}{partial x_{n}}(p)
end{pmatrix}.
$$

By assumption, this matrix has rank $r$, so there is a nonzero minor of size $rtimes r$:
$$
begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(p)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(p)
end{vmatrix}neq 0.
$$
The map
$$
G:qmapsto begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(q)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(q)
end{vmatrix},
$$
is continuous and nonzero at $p$, so there exists a neighborhood $U$ of $p$ such that $G(q)neq 0$ for all $qin U$. This implies that for all $qin U$
$$
d_{q}F=begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(q) & cdots & frac{partial F_{1}}{partial x_{n}}(q)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(q) & cdots & frac{partial F_{m}}{partial x_{n}}(q)
end{pmatrix}
$$
has rank at least $r$, since it has a nonzero $rtimes r$ minor.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Very clear. Really appreciate:)
    $endgroup$
    – user450201
    Dec 20 '18 at 11:11














1












1








1





$begingroup$

Let us choose coordinate charts $(x_{1},ldots,x_{n})$ around $p$ and $(y_{1},ldots,y_{m})$ around $F(p)$. Writing $F=big(F_{1}(x_{1},ldots,x_{n}),ldots,F_{m}(x_{1},ldots,x_{n})big)$, we can express the derivative $d_{p}F$ as a matrix
$$
begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(p) & cdots & frac{partial F_{1}}{partial x_{n}}(p)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(p) & cdots & frac{partial F_{m}}{partial x_{n}}(p)
end{pmatrix}.
$$

By assumption, this matrix has rank $r$, so there is a nonzero minor of size $rtimes r$:
$$
begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(p)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(p)
end{vmatrix}neq 0.
$$
The map
$$
G:qmapsto begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(q)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(q)
end{vmatrix},
$$
is continuous and nonzero at $p$, so there exists a neighborhood $U$ of $p$ such that $G(q)neq 0$ for all $qin U$. This implies that for all $qin U$
$$
d_{q}F=begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(q) & cdots & frac{partial F_{1}}{partial x_{n}}(q)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(q) & cdots & frac{partial F_{m}}{partial x_{n}}(q)
end{pmatrix}
$$
has rank at least $r$, since it has a nonzero $rtimes r$ minor.






share|cite|improve this answer









$endgroup$



Let us choose coordinate charts $(x_{1},ldots,x_{n})$ around $p$ and $(y_{1},ldots,y_{m})$ around $F(p)$. Writing $F=big(F_{1}(x_{1},ldots,x_{n}),ldots,F_{m}(x_{1},ldots,x_{n})big)$, we can express the derivative $d_{p}F$ as a matrix
$$
begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(p) & cdots & frac{partial F_{1}}{partial x_{n}}(p)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(p) & cdots & frac{partial F_{m}}{partial x_{n}}(p)
end{pmatrix}.
$$

By assumption, this matrix has rank $r$, so there is a nonzero minor of size $rtimes r$:
$$
begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(p)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(p) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(p)
end{vmatrix}neq 0.
$$
The map
$$
G:qmapsto begin{vmatrix}
frac{partial F_{i_{1}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{1}}}{partial x_{j_{r}}}(q)\
vdots & & vdots\
frac{partial F_{i_{r}}}{partial x_{j_{1}}}(q) & cdots & frac{partial F_{i_{r}}}{partial x_{j_{r}}}(q)
end{vmatrix},
$$
is continuous and nonzero at $p$, so there exists a neighborhood $U$ of $p$ such that $G(q)neq 0$ for all $qin U$. This implies that for all $qin U$
$$
d_{q}F=begin{pmatrix}
frac{partial F_{1}}{partial x_{1}}(q) & cdots & frac{partial F_{1}}{partial x_{n}}(q)\
vdots & & vdots\
frac{partial F_{m}}{partial x_{1}}(q) & cdots & frac{partial F_{m}}{partial x_{n}}(q)
end{pmatrix}
$$
has rank at least $r$, since it has a nonzero $rtimes r$ minor.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 20 '18 at 11:06









studiosusstudiosus

1,807713




1,807713












  • $begingroup$
    Very clear. Really appreciate:)
    $endgroup$
    – user450201
    Dec 20 '18 at 11:11


















  • $begingroup$
    Very clear. Really appreciate:)
    $endgroup$
    – user450201
    Dec 20 '18 at 11:11
















$begingroup$
Very clear. Really appreciate:)
$endgroup$
– user450201
Dec 20 '18 at 11:11




$begingroup$
Very clear. Really appreciate:)
$endgroup$
– user450201
Dec 20 '18 at 11:11


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3046105%2frank-of-a-smooth-map-is-lower-semicontinuous%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Bressuire

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more

Cabo Verde

Image
  Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
Read more

Gyllenstierna

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more