A vector field corresponding to the complement of the tangent bundle












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Let $M$ be a $m$ dimensional orientable manifold, and $N$ a $m-1$ dimensional orientable submanifold in $M$, then we know at each point $x in N$, $T_{x}M = T_x N oplus$its complement. I need to produce a vector field $X$ such that $X_x$ is a vector in the complement of $T_{x}N$.



I am trying to exploit the assumption that both manifolds are orientable meaning they all have a non-vanishing top form, but what next?



UPDATE1:
Take a slice chart $(x_1, ldots, x_n)$ of $M$. Let the orientation form on $M$ be $w_M = fdx_1 wedge ldots wedge dx_n$, and the orientation form on $N$ be $w_N = fdx_1 wedge ldots wedge dx_{n-1}$. A possible approach is to find a vector $V$ such that $V lnot w_{M} = w_N$. The computation is easy. Locally $V$ should be $frac{g}{f}frac{partial}{partial x_n}$. However, how to show this expression is independent of coordinate thus can be globalized?



UPDATE2:
Trying to understand @Tsemo Aristide's answer. Locally, the choice of the sign has to be consistent. so in a neighborhood, it has to be either $u_x$ or $-u_x$ throughout, which proves the smoothness. Is this correct?










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    1












    $begingroup$


    Let $M$ be a $m$ dimensional orientable manifold, and $N$ a $m-1$ dimensional orientable submanifold in $M$, then we know at each point $x in N$, $T_{x}M = T_x N oplus$its complement. I need to produce a vector field $X$ such that $X_x$ is a vector in the complement of $T_{x}N$.



    I am trying to exploit the assumption that both manifolds are orientable meaning they all have a non-vanishing top form, but what next?



    UPDATE1:
    Take a slice chart $(x_1, ldots, x_n)$ of $M$. Let the orientation form on $M$ be $w_M = fdx_1 wedge ldots wedge dx_n$, and the orientation form on $N$ be $w_N = fdx_1 wedge ldots wedge dx_{n-1}$. A possible approach is to find a vector $V$ such that $V lnot w_{M} = w_N$. The computation is easy. Locally $V$ should be $frac{g}{f}frac{partial}{partial x_n}$. However, how to show this expression is independent of coordinate thus can be globalized?



    UPDATE2:
    Trying to understand @Tsemo Aristide's answer. Locally, the choice of the sign has to be consistent. so in a neighborhood, it has to be either $u_x$ or $-u_x$ throughout, which proves the smoothness. Is this correct?










    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      Let $M$ be a $m$ dimensional orientable manifold, and $N$ a $m-1$ dimensional orientable submanifold in $M$, then we know at each point $x in N$, $T_{x}M = T_x N oplus$its complement. I need to produce a vector field $X$ such that $X_x$ is a vector in the complement of $T_{x}N$.



      I am trying to exploit the assumption that both manifolds are orientable meaning they all have a non-vanishing top form, but what next?



      UPDATE1:
      Take a slice chart $(x_1, ldots, x_n)$ of $M$. Let the orientation form on $M$ be $w_M = fdx_1 wedge ldots wedge dx_n$, and the orientation form on $N$ be $w_N = fdx_1 wedge ldots wedge dx_{n-1}$. A possible approach is to find a vector $V$ such that $V lnot w_{M} = w_N$. The computation is easy. Locally $V$ should be $frac{g}{f}frac{partial}{partial x_n}$. However, how to show this expression is independent of coordinate thus can be globalized?



      UPDATE2:
      Trying to understand @Tsemo Aristide's answer. Locally, the choice of the sign has to be consistent. so in a neighborhood, it has to be either $u_x$ or $-u_x$ throughout, which proves the smoothness. Is this correct?










      share|cite|improve this question











      $endgroup$




      Let $M$ be a $m$ dimensional orientable manifold, and $N$ a $m-1$ dimensional orientable submanifold in $M$, then we know at each point $x in N$, $T_{x}M = T_x N oplus$its complement. I need to produce a vector field $X$ such that $X_x$ is a vector in the complement of $T_{x}N$.



      I am trying to exploit the assumption that both manifolds are orientable meaning they all have a non-vanishing top form, but what next?



      UPDATE1:
      Take a slice chart $(x_1, ldots, x_n)$ of $M$. Let the orientation form on $M$ be $w_M = fdx_1 wedge ldots wedge dx_n$, and the orientation form on $N$ be $w_N = fdx_1 wedge ldots wedge dx_{n-1}$. A possible approach is to find a vector $V$ such that $V lnot w_{M} = w_N$. The computation is easy. Locally $V$ should be $frac{g}{f}frac{partial}{partial x_n}$. However, how to show this expression is independent of coordinate thus can be globalized?



      UPDATE2:
      Trying to understand @Tsemo Aristide's answer. Locally, the choice of the sign has to be consistent. so in a neighborhood, it has to be either $u_x$ or $-u_x$ throughout, which proves the smoothness. Is this correct?







      differential-topology






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      edited Dec 10 '18 at 5:02







      Keith

















      asked Dec 7 '18 at 23:39









      KeithKeith

      420317




      420317






















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          $begingroup$

          Let $Omega^M$ be the volume form of $M$ and $Omega^N$ the volume form of $N$. Consider a
          differentiable metric defined on $M$. For every $xin N$, there exists two vectors of norm $1$, $u_x,-u_x$ orthogonal to $T_xN$, If the restriction of $i_{u_x}{Omega^M}_x$ to $T_xN$ is $c{Omega^N}_x, c>0$ write $n(x)=u_x$ otherwise, write $n(x)=-u_x$. Show that $n(x)$ is differentiable by using local coordinates.






          share|cite|improve this answer









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          • $begingroup$
            May you be more specific about how to check the differentiability? I cannot really figure out how to write down an explicit formula locally
            $endgroup$
            – Keith
            Dec 8 '18 at 2:17













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          $begingroup$

          Let $Omega^M$ be the volume form of $M$ and $Omega^N$ the volume form of $N$. Consider a
          differentiable metric defined on $M$. For every $xin N$, there exists two vectors of norm $1$, $u_x,-u_x$ orthogonal to $T_xN$, If the restriction of $i_{u_x}{Omega^M}_x$ to $T_xN$ is $c{Omega^N}_x, c>0$ write $n(x)=u_x$ otherwise, write $n(x)=-u_x$. Show that $n(x)$ is differentiable by using local coordinates.






          share|cite|improve this answer









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          • $begingroup$
            May you be more specific about how to check the differentiability? I cannot really figure out how to write down an explicit formula locally
            $endgroup$
            – Keith
            Dec 8 '18 at 2:17


















          2












          $begingroup$

          Let $Omega^M$ be the volume form of $M$ and $Omega^N$ the volume form of $N$. Consider a
          differentiable metric defined on $M$. For every $xin N$, there exists two vectors of norm $1$, $u_x,-u_x$ orthogonal to $T_xN$, If the restriction of $i_{u_x}{Omega^M}_x$ to $T_xN$ is $c{Omega^N}_x, c>0$ write $n(x)=u_x$ otherwise, write $n(x)=-u_x$. Show that $n(x)$ is differentiable by using local coordinates.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            May you be more specific about how to check the differentiability? I cannot really figure out how to write down an explicit formula locally
            $endgroup$
            – Keith
            Dec 8 '18 at 2:17
















          2












          2








          2





          $begingroup$

          Let $Omega^M$ be the volume form of $M$ and $Omega^N$ the volume form of $N$. Consider a
          differentiable metric defined on $M$. For every $xin N$, there exists two vectors of norm $1$, $u_x,-u_x$ orthogonal to $T_xN$, If the restriction of $i_{u_x}{Omega^M}_x$ to $T_xN$ is $c{Omega^N}_x, c>0$ write $n(x)=u_x$ otherwise, write $n(x)=-u_x$. Show that $n(x)$ is differentiable by using local coordinates.






          share|cite|improve this answer









          $endgroup$



          Let $Omega^M$ be the volume form of $M$ and $Omega^N$ the volume form of $N$. Consider a
          differentiable metric defined on $M$. For every $xin N$, there exists two vectors of norm $1$, $u_x,-u_x$ orthogonal to $T_xN$, If the restriction of $i_{u_x}{Omega^M}_x$ to $T_xN$ is $c{Omega^N}_x, c>0$ write $n(x)=u_x$ otherwise, write $n(x)=-u_x$. Show that $n(x)$ is differentiable by using local coordinates.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 8 '18 at 0:25









          Tsemo AristideTsemo Aristide

          57.5k11444




          57.5k11444












          • $begingroup$
            May you be more specific about how to check the differentiability? I cannot really figure out how to write down an explicit formula locally
            $endgroup$
            – Keith
            Dec 8 '18 at 2:17




















          • $begingroup$
            May you be more specific about how to check the differentiability? I cannot really figure out how to write down an explicit formula locally
            $endgroup$
            – Keith
            Dec 8 '18 at 2:17


















          $begingroup$
          May you be more specific about how to check the differentiability? I cannot really figure out how to write down an explicit formula locally
          $endgroup$
          – Keith
          Dec 8 '18 at 2:17






          $begingroup$
          May you be more specific about how to check the differentiability? I cannot really figure out how to write down an explicit formula locally
          $endgroup$
          – Keith
          Dec 8 '18 at 2:17




















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