Understanding the definitions of Embedded Surface and Locally Parametrised Embedded Surface











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I recently learned about the definitions of Embedded Surface (ES) and Locally Parametrised Embedded Surface (LEPS) in a lecture of mine. We defined them as follows:



ES: A regular parametrisation $f$ is called an Embedded Surface if for $f:D rightarrow f(D) = S subseteq mathbb{R}^n$ holds:




  • For all $uin D$ and for all open $V subseteq D$ with $u in V$ there exists an open $U subseteq mathbb{R}^n$ with $f(u) in U$ such that $U cap S = f(V)$.


LPES: We say $S subseteq mathbb{R}^n$ is a Locally Parametrised Embedded (regular)($k$-dimensional) Surface if for all $x in S$ there exists an open neighbourhood $U subseteq mathbb{R}^n$ of $x$ in $mathbb{R}^n$ and a domain $D subseteq mathbb{R}^k$ as well as a regular parametrisation $f:Drightarrow mathbb{R}^n$ sucht that
$$U cap S = f(D)$$



Regular Parametrisation: Let $D subseteq mathbb{R}^k$ be a domain an $n,k in mathbb{N}$ such that $n ge k$. Now consider the injective function $f: D rightarrow mathbb{R}^n in C^1(D,mathbb{R}^n)$ and assume that it's Jacobian has rank $k$. We call $f$ a regular parametrisation (of a k-dimensional surface in $mathbb{R}^n$).



I got the following questions:




  1. If I understand these definitions correctly then every ES is also an LPES but the other direction does not hold since an ES requires that $f$ is alwys the same parametrisation for every $x in S$. Is this rigth?

  2. Are there any intuitive examples of an LPES that is not an ES? I can hardly imagine something such as an LPES in terms of "real-world" geometry.










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    I recently learned about the definitions of Embedded Surface (ES) and Locally Parametrised Embedded Surface (LEPS) in a lecture of mine. We defined them as follows:



    ES: A regular parametrisation $f$ is called an Embedded Surface if for $f:D rightarrow f(D) = S subseteq mathbb{R}^n$ holds:




    • For all $uin D$ and for all open $V subseteq D$ with $u in V$ there exists an open $U subseteq mathbb{R}^n$ with $f(u) in U$ such that $U cap S = f(V)$.


    LPES: We say $S subseteq mathbb{R}^n$ is a Locally Parametrised Embedded (regular)($k$-dimensional) Surface if for all $x in S$ there exists an open neighbourhood $U subseteq mathbb{R}^n$ of $x$ in $mathbb{R}^n$ and a domain $D subseteq mathbb{R}^k$ as well as a regular parametrisation $f:Drightarrow mathbb{R}^n$ sucht that
    $$U cap S = f(D)$$



    Regular Parametrisation: Let $D subseteq mathbb{R}^k$ be a domain an $n,k in mathbb{N}$ such that $n ge k$. Now consider the injective function $f: D rightarrow mathbb{R}^n in C^1(D,mathbb{R}^n)$ and assume that it's Jacobian has rank $k$. We call $f$ a regular parametrisation (of a k-dimensional surface in $mathbb{R}^n$).



    I got the following questions:




    1. If I understand these definitions correctly then every ES is also an LPES but the other direction does not hold since an ES requires that $f$ is alwys the same parametrisation for every $x in S$. Is this rigth?

    2. Are there any intuitive examples of an LPES that is not an ES? I can hardly imagine something such as an LPES in terms of "real-world" geometry.










    share|cite|improve this question


























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I recently learned about the definitions of Embedded Surface (ES) and Locally Parametrised Embedded Surface (LEPS) in a lecture of mine. We defined them as follows:



      ES: A regular parametrisation $f$ is called an Embedded Surface if for $f:D rightarrow f(D) = S subseteq mathbb{R}^n$ holds:




      • For all $uin D$ and for all open $V subseteq D$ with $u in V$ there exists an open $U subseteq mathbb{R}^n$ with $f(u) in U$ such that $U cap S = f(V)$.


      LPES: We say $S subseteq mathbb{R}^n$ is a Locally Parametrised Embedded (regular)($k$-dimensional) Surface if for all $x in S$ there exists an open neighbourhood $U subseteq mathbb{R}^n$ of $x$ in $mathbb{R}^n$ and a domain $D subseteq mathbb{R}^k$ as well as a regular parametrisation $f:Drightarrow mathbb{R}^n$ sucht that
      $$U cap S = f(D)$$



      Regular Parametrisation: Let $D subseteq mathbb{R}^k$ be a domain an $n,k in mathbb{N}$ such that $n ge k$. Now consider the injective function $f: D rightarrow mathbb{R}^n in C^1(D,mathbb{R}^n)$ and assume that it's Jacobian has rank $k$. We call $f$ a regular parametrisation (of a k-dimensional surface in $mathbb{R}^n$).



      I got the following questions:




      1. If I understand these definitions correctly then every ES is also an LPES but the other direction does not hold since an ES requires that $f$ is alwys the same parametrisation for every $x in S$. Is this rigth?

      2. Are there any intuitive examples of an LPES that is not an ES? I can hardly imagine something such as an LPES in terms of "real-world" geometry.










      share|cite|improve this question















      I recently learned about the definitions of Embedded Surface (ES) and Locally Parametrised Embedded Surface (LEPS) in a lecture of mine. We defined them as follows:



      ES: A regular parametrisation $f$ is called an Embedded Surface if for $f:D rightarrow f(D) = S subseteq mathbb{R}^n$ holds:




      • For all $uin D$ and for all open $V subseteq D$ with $u in V$ there exists an open $U subseteq mathbb{R}^n$ with $f(u) in U$ such that $U cap S = f(V)$.


      LPES: We say $S subseteq mathbb{R}^n$ is a Locally Parametrised Embedded (regular)($k$-dimensional) Surface if for all $x in S$ there exists an open neighbourhood $U subseteq mathbb{R}^n$ of $x$ in $mathbb{R}^n$ and a domain $D subseteq mathbb{R}^k$ as well as a regular parametrisation $f:Drightarrow mathbb{R}^n$ sucht that
      $$U cap S = f(D)$$



      Regular Parametrisation: Let $D subseteq mathbb{R}^k$ be a domain an $n,k in mathbb{N}$ such that $n ge k$. Now consider the injective function $f: D rightarrow mathbb{R}^n in C^1(D,mathbb{R}^n)$ and assume that it's Jacobian has rank $k$. We call $f$ a regular parametrisation (of a k-dimensional surface in $mathbb{R}^n$).



      I got the following questions:




      1. If I understand these definitions correctly then every ES is also an LPES but the other direction does not hold since an ES requires that $f$ is alwys the same parametrisation for every $x in S$. Is this rigth?

      2. Are there any intuitive examples of an LPES that is not an ES? I can hardly imagine something such as an LPES in terms of "real-world" geometry.







      calculus differential-geometry manifolds surfaces






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      edited Nov 13 at 22:23

























      asked Nov 13 at 21:50









      3nondatur

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          Before answering the question, I want to point out that these terms are not standard. The standard terms for "regular parametrization," "embedded surface," and "locally parametrized embedded surface" are, respectively: "injective immersion," "embedding," and "embedded submanifold."



          As to your question: Technically speaking, according to the definitions you've given: An ES refers to a special kind of function. That is, an ES is a regular parametrization that satisfies a nice property. By contrast, an LPES is a special kind of subset of $mathbb{R}^n$.



          An LPES is a very concrete, geometric object. Here's an example:




          Example: The unit sphere $mathbb{S} = {(x,y,z) in mathbb{R}^3 colon x^2 + y^2 + z^2 = 1}$ is an LPES. Indeed, for any point $(x,y,z) in mathbb{S}$ on the sphere, you can find an open set $U subset mathbb{R}^3$ containing $(x,y,z)$ for which $U cap mathbb{S} = f(D)$ for some domain $D subset mathbb{R}^2$ and some regular parametrization $f colon D to mathbb{R}^3$.



          So, let's say we take $(x,y,z) = (0,0,1)$. Then the upper half-space $U = {(x,y,z) colon z > 0}$ is an open subset in $mathbb{R}^3$ with $(x,y,z) in U$, and the intersection $U cap mathbb{S}$ is the upper hemisphere. You can then take $D = {(u,v) in mathbb{R}^2 colon u^2 + v^2 = 1}$ to be the unit disk in $mathbb{R}^2$ and take $f colon D to mathbb{R}^3$ to be $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$.



          One of the things to take away from this example is that the regular parametrization $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$ only gives you points on the upper hemisphere. You will need a different regular parametrization to get points on the lower hemisphere. Even then you haven't covered the entire sphere: the points on the equator will require (at least) one additional regular parametrization.







          share|cite|improve this answer





















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            1 Answer
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            active

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            active

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            up vote
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            accepted










            Before answering the question, I want to point out that these terms are not standard. The standard terms for "regular parametrization," "embedded surface," and "locally parametrized embedded surface" are, respectively: "injective immersion," "embedding," and "embedded submanifold."



            As to your question: Technically speaking, according to the definitions you've given: An ES refers to a special kind of function. That is, an ES is a regular parametrization that satisfies a nice property. By contrast, an LPES is a special kind of subset of $mathbb{R}^n$.



            An LPES is a very concrete, geometric object. Here's an example:




            Example: The unit sphere $mathbb{S} = {(x,y,z) in mathbb{R}^3 colon x^2 + y^2 + z^2 = 1}$ is an LPES. Indeed, for any point $(x,y,z) in mathbb{S}$ on the sphere, you can find an open set $U subset mathbb{R}^3$ containing $(x,y,z)$ for which $U cap mathbb{S} = f(D)$ for some domain $D subset mathbb{R}^2$ and some regular parametrization $f colon D to mathbb{R}^3$.



            So, let's say we take $(x,y,z) = (0,0,1)$. Then the upper half-space $U = {(x,y,z) colon z > 0}$ is an open subset in $mathbb{R}^3$ with $(x,y,z) in U$, and the intersection $U cap mathbb{S}$ is the upper hemisphere. You can then take $D = {(u,v) in mathbb{R}^2 colon u^2 + v^2 = 1}$ to be the unit disk in $mathbb{R}^2$ and take $f colon D to mathbb{R}^3$ to be $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$.



            One of the things to take away from this example is that the regular parametrization $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$ only gives you points on the upper hemisphere. You will need a different regular parametrization to get points on the lower hemisphere. Even then you haven't covered the entire sphere: the points on the equator will require (at least) one additional regular parametrization.







            share|cite|improve this answer

























              up vote
              1
              down vote



              accepted










              Before answering the question, I want to point out that these terms are not standard. The standard terms for "regular parametrization," "embedded surface," and "locally parametrized embedded surface" are, respectively: "injective immersion," "embedding," and "embedded submanifold."



              As to your question: Technically speaking, according to the definitions you've given: An ES refers to a special kind of function. That is, an ES is a regular parametrization that satisfies a nice property. By contrast, an LPES is a special kind of subset of $mathbb{R}^n$.



              An LPES is a very concrete, geometric object. Here's an example:




              Example: The unit sphere $mathbb{S} = {(x,y,z) in mathbb{R}^3 colon x^2 + y^2 + z^2 = 1}$ is an LPES. Indeed, for any point $(x,y,z) in mathbb{S}$ on the sphere, you can find an open set $U subset mathbb{R}^3$ containing $(x,y,z)$ for which $U cap mathbb{S} = f(D)$ for some domain $D subset mathbb{R}^2$ and some regular parametrization $f colon D to mathbb{R}^3$.



              So, let's say we take $(x,y,z) = (0,0,1)$. Then the upper half-space $U = {(x,y,z) colon z > 0}$ is an open subset in $mathbb{R}^3$ with $(x,y,z) in U$, and the intersection $U cap mathbb{S}$ is the upper hemisphere. You can then take $D = {(u,v) in mathbb{R}^2 colon u^2 + v^2 = 1}$ to be the unit disk in $mathbb{R}^2$ and take $f colon D to mathbb{R}^3$ to be $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$.



              One of the things to take away from this example is that the regular parametrization $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$ only gives you points on the upper hemisphere. You will need a different regular parametrization to get points on the lower hemisphere. Even then you haven't covered the entire sphere: the points on the equator will require (at least) one additional regular parametrization.







              share|cite|improve this answer























                up vote
                1
                down vote



                accepted







                up vote
                1
                down vote



                accepted






                Before answering the question, I want to point out that these terms are not standard. The standard terms for "regular parametrization," "embedded surface," and "locally parametrized embedded surface" are, respectively: "injective immersion," "embedding," and "embedded submanifold."



                As to your question: Technically speaking, according to the definitions you've given: An ES refers to a special kind of function. That is, an ES is a regular parametrization that satisfies a nice property. By contrast, an LPES is a special kind of subset of $mathbb{R}^n$.



                An LPES is a very concrete, geometric object. Here's an example:




                Example: The unit sphere $mathbb{S} = {(x,y,z) in mathbb{R}^3 colon x^2 + y^2 + z^2 = 1}$ is an LPES. Indeed, for any point $(x,y,z) in mathbb{S}$ on the sphere, you can find an open set $U subset mathbb{R}^3$ containing $(x,y,z)$ for which $U cap mathbb{S} = f(D)$ for some domain $D subset mathbb{R}^2$ and some regular parametrization $f colon D to mathbb{R}^3$.



                So, let's say we take $(x,y,z) = (0,0,1)$. Then the upper half-space $U = {(x,y,z) colon z > 0}$ is an open subset in $mathbb{R}^3$ with $(x,y,z) in U$, and the intersection $U cap mathbb{S}$ is the upper hemisphere. You can then take $D = {(u,v) in mathbb{R}^2 colon u^2 + v^2 = 1}$ to be the unit disk in $mathbb{R}^2$ and take $f colon D to mathbb{R}^3$ to be $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$.



                One of the things to take away from this example is that the regular parametrization $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$ only gives you points on the upper hemisphere. You will need a different regular parametrization to get points on the lower hemisphere. Even then you haven't covered the entire sphere: the points on the equator will require (at least) one additional regular parametrization.







                share|cite|improve this answer












                Before answering the question, I want to point out that these terms are not standard. The standard terms for "regular parametrization," "embedded surface," and "locally parametrized embedded surface" are, respectively: "injective immersion," "embedding," and "embedded submanifold."



                As to your question: Technically speaking, according to the definitions you've given: An ES refers to a special kind of function. That is, an ES is a regular parametrization that satisfies a nice property. By contrast, an LPES is a special kind of subset of $mathbb{R}^n$.



                An LPES is a very concrete, geometric object. Here's an example:




                Example: The unit sphere $mathbb{S} = {(x,y,z) in mathbb{R}^3 colon x^2 + y^2 + z^2 = 1}$ is an LPES. Indeed, for any point $(x,y,z) in mathbb{S}$ on the sphere, you can find an open set $U subset mathbb{R}^3$ containing $(x,y,z)$ for which $U cap mathbb{S} = f(D)$ for some domain $D subset mathbb{R}^2$ and some regular parametrization $f colon D to mathbb{R}^3$.



                So, let's say we take $(x,y,z) = (0,0,1)$. Then the upper half-space $U = {(x,y,z) colon z > 0}$ is an open subset in $mathbb{R}^3$ with $(x,y,z) in U$, and the intersection $U cap mathbb{S}$ is the upper hemisphere. You can then take $D = {(u,v) in mathbb{R}^2 colon u^2 + v^2 = 1}$ to be the unit disk in $mathbb{R}^2$ and take $f colon D to mathbb{R}^3$ to be $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$.



                One of the things to take away from this example is that the regular parametrization $f(u,v) = (u,v, sqrt{1 - u^2 - v^2})$ only gives you points on the upper hemisphere. You will need a different regular parametrization to get points on the lower hemisphere. Even then you haven't covered the entire sphere: the points on the equator will require (at least) one additional regular parametrization.








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                answered Nov 14 at 0:30









                Jesse Madnick

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