Definition of manifolds as submanifolds of $mathbb{R}^m$











up vote
2
down vote

favorite












I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $mathbb{R}^m$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).



The definition of a submanifold I learned is:




Let $M$ be an $m$-dimensional manifold. We say that a subset $L subseteq M$ is a submanifold of $M$ of dimension $n$, if for every point $p in L$ there exists an adapted chart $phi: U rightarrow V' times V''$ with $U subseteq L$ open in $L$, $V' subseteq mathbb{R}^n$ open in $mathbb{R}^n$ and $V'' subseteq mathbb{R}^{m-n}$ open in $mathbb{R}^{m-n}$ such that $phi (U cap L) = V' times {0}$ with $0 in mathbb{R}^{m-n}$ Then $L$ becomes an $n$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $phi : U cap L rightarrow V'$.




The authors of the book define a submanifold of $mathbb{R}^m$ as follows




A subset $L subseteq mathbb{R}^m$ is called an $n$-dimensional submanifold of $mathbb{R}^m$ if for every point $p in L$ there exists an open set $U subseteq mathbb{R}^m$ containing $p$ and an open subset $Vsubseteq mathbb{R}^m$ together with a diffeomorphism $phi$ from $U$ to $V$ such that $phi(M cap U)=V cap (mathbb{R}^n times {0})$ with $0 in mathbb{R}^{m-n}$.




,which directly coincides with the book's definition if one views $mathbb{R}^m$ as an $m$-dimensional manifold with the smooth atlas induced by the universal chart $(mathbb{R}^m, id)$.
Then the authors define coordinate charts of an $n$-dimensional submanifold $L$ of $mathbb{R}^m$ around a point $p in L$ as follows




Let $phi: U rightarrow V$ be a homeomorphism from an open set $U subseteq L$ in the subspace topology containing $p$ to an open subset $V$ of $mathbb{R}^n$ such that $ i_M circ phi ^{-1}$ is a $C^infty$-immersion, where $i_M$ is the cannonical injection from $L$ to $mathbb{R}^m$.




How is this equivalent to the restricted charts given in the first definition? Why does $ i_M circ phi^{-1}$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $mathbb{R}^m$. What's a good way to think about the approach taken by the book?






differential-geometry definition smooth-manifolds







share|cite|improve this question


























    up vote
    2
    down vote

    favorite












    I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $mathbb{R}^m$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).



    The definition of a submanifold I learned is:




    Let $M$ be an $m$-dimensional manifold. We say that a subset $L subseteq M$ is a submanifold of $M$ of dimension $n$, if for every point $p in L$ there exists an adapted chart $phi: U rightarrow V' times V''$ with $U subseteq L$ open in $L$, $V' subseteq mathbb{R}^n$ open in $mathbb{R}^n$ and $V'' subseteq mathbb{R}^{m-n}$ open in $mathbb{R}^{m-n}$ such that $phi (U cap L) = V' times {0}$ with $0 in mathbb{R}^{m-n}$ Then $L$ becomes an $n$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $phi : U cap L rightarrow V'$.




    The authors of the book define a submanifold of $mathbb{R}^m$ as follows




    A subset $L subseteq mathbb{R}^m$ is called an $n$-dimensional submanifold of $mathbb{R}^m$ if for every point $p in L$ there exists an open set $U subseteq mathbb{R}^m$ containing $p$ and an open subset $Vsubseteq mathbb{R}^m$ together with a diffeomorphism $phi$ from $U$ to $V$ such that $phi(M cap U)=V cap (mathbb{R}^n times {0})$ with $0 in mathbb{R}^{m-n}$.




    ,which directly coincides with the book's definition if one views $mathbb{R}^m$ as an $m$-dimensional manifold with the smooth atlas induced by the universal chart $(mathbb{R}^m, id)$.
    Then the authors define coordinate charts of an $n$-dimensional submanifold $L$ of $mathbb{R}^m$ around a point $p in L$ as follows




    Let $phi: U rightarrow V$ be a homeomorphism from an open set $U subseteq L$ in the subspace topology containing $p$ to an open subset $V$ of $mathbb{R}^n$ such that $ i_M circ phi ^{-1}$ is a $C^infty$-immersion, where $i_M$ is the cannonical injection from $L$ to $mathbb{R}^m$.




    How is this equivalent to the restricted charts given in the first definition? Why does $ i_M circ phi^{-1}$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $mathbb{R}^m$. What's a good way to think about the approach taken by the book?






    differential-geometry definition smooth-manifolds







    share|cite|improve this question
























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $mathbb{R}^m$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).



      The definition of a submanifold I learned is:




      Let $M$ be an $m$-dimensional manifold. We say that a subset $L subseteq M$ is a submanifold of $M$ of dimension $n$, if for every point $p in L$ there exists an adapted chart $phi: U rightarrow V' times V''$ with $U subseteq L$ open in $L$, $V' subseteq mathbb{R}^n$ open in $mathbb{R}^n$ and $V'' subseteq mathbb{R}^{m-n}$ open in $mathbb{R}^{m-n}$ such that $phi (U cap L) = V' times {0}$ with $0 in mathbb{R}^{m-n}$ Then $L$ becomes an $n$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $phi : U cap L rightarrow V'$.




      The authors of the book define a submanifold of $mathbb{R}^m$ as follows




      A subset $L subseteq mathbb{R}^m$ is called an $n$-dimensional submanifold of $mathbb{R}^m$ if for every point $p in L$ there exists an open set $U subseteq mathbb{R}^m$ containing $p$ and an open subset $Vsubseteq mathbb{R}^m$ together with a diffeomorphism $phi$ from $U$ to $V$ such that $phi(M cap U)=V cap (mathbb{R}^n times {0})$ with $0 in mathbb{R}^{m-n}$.




      ,which directly coincides with the book's definition if one views $mathbb{R}^m$ as an $m$-dimensional manifold with the smooth atlas induced by the universal chart $(mathbb{R}^m, id)$.
      Then the authors define coordinate charts of an $n$-dimensional submanifold $L$ of $mathbb{R}^m$ around a point $p in L$ as follows




      Let $phi: U rightarrow V$ be a homeomorphism from an open set $U subseteq L$ in the subspace topology containing $p$ to an open subset $V$ of $mathbb{R}^n$ such that $ i_M circ phi ^{-1}$ is a $C^infty$-immersion, where $i_M$ is the cannonical injection from $L$ to $mathbb{R}^m$.




      How is this equivalent to the restricted charts given in the first definition? Why does $ i_M circ phi^{-1}$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $mathbb{R}^m$. What's a good way to think about the approach taken by the book?






      differential-geometry definition smooth-manifolds







      share|cite|improve this question













      I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $mathbb{R}^m$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).



      The definition of a submanifold I learned is:




      Let $M$ be an $m$-dimensional manifold. We say that a subset $L subseteq M$ is a submanifold of $M$ of dimension $n$, if for every point $p in L$ there exists an adapted chart $phi: U rightarrow V' times V''$ with $U subseteq L$ open in $L$, $V' subseteq mathbb{R}^n$ open in $mathbb{R}^n$ and $V'' subseteq mathbb{R}^{m-n}$ open in $mathbb{R}^{m-n}$ such that $phi (U cap L) = V' times {0}$ with $0 in mathbb{R}^{m-n}$ Then $L$ becomes an $n$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $phi : U cap L rightarrow V'$.




      The authors of the book define a submanifold of $mathbb{R}^m$ as follows




      A subset $L subseteq mathbb{R}^m$ is called an $n$-dimensional submanifold of $mathbb{R}^m$ if for every point $p in L$ there exists an open set $U subseteq mathbb{R}^m$ containing $p$ and an open subset $Vsubseteq mathbb{R}^m$ together with a diffeomorphism $phi$ from $U$ to $V$ such that $phi(M cap U)=V cap (mathbb{R}^n times {0})$ with $0 in mathbb{R}^{m-n}$.




      ,which directly coincides with the book's definition if one views $mathbb{R}^m$ as an $m$-dimensional manifold with the smooth atlas induced by the universal chart $(mathbb{R}^m, id)$.
      Then the authors define coordinate charts of an $n$-dimensional submanifold $L$ of $mathbb{R}^m$ around a point $p in L$ as follows




      Let $phi: U rightarrow V$ be a homeomorphism from an open set $U subseteq L$ in the subspace topology containing $p$ to an open subset $V$ of $mathbb{R}^n$ such that $ i_M circ phi ^{-1}$ is a $C^infty$-immersion, where $i_M$ is the cannonical injection from $L$ to $mathbb{R}^m$.




      How is this equivalent to the restricted charts given in the first definition? Why does $ i_M circ phi^{-1}$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $mathbb{R}^m$. What's a good way to think about the approach taken by the book?






      differential-geometry definition smooth-manifolds




      differential-geometry definition smooth-manifolds






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 22 at 20:03









      Jannik Pitt

      291316




      291316



























          active

          oldest

          votes











          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',
          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%2f3009584%2fdefinition-of-manifolds-as-submanifolds-of-mathbbrm%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown






























          active

          oldest

          votes













          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          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.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • 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.


          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%2f3009584%2fdefinition-of-manifolds-as-submanifolds-of-mathbbrm%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

          Ellipse (mathématiques)

          Image
          Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
          Read more

          Quarter-circle Tiles

          Image
          up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
          Read more

          Mont Emei

          Image
          Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
          Read more