Results 1 to 4 of 4

Thread: Bijection between tangent spaces

  1. #1
    Member
    Joined
    Oct 2010
    Posts
    131

    Bijection between tangent spaces

    Hello,

    We have defined the tangent space at a point p of a manifold as the set of all derivatives.
    Now i want to show, that there exist a bijection between all derivatives and the "Tangentvectors" defined by (equivalent classes of) paths.
    More precisely:
    We call two paths equivalent, iff they define the same derivation at a point p. Now i want to show that there is a bijection between the set of all equivalent classes and the Tangentspace (defined as the set of derivatives).

    My Approach was the following.
    Every path c:I->M into the manifold M defines a derivation by
    $\displaystyle D(f):=\frac{d}{dt}_{|t=0}(f\circ c)(t)$

    so i try to show that $\displaystyle \phi([c])=D$ with $\displaystyle D(f)=\frac{d}{dt}_{|t=0}(f\circ c)(t)$ is a bijection.

    Of course $\displaystyle \phi$ is well defined and injective, but i couldn't show that
    $\displaystyle \phi$ is surjective!

    If D is some arbitrary derivative at the point p. How can i find a path c, s.t. $\displaystyle \phi(c)=D$?

    Regards
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    My house.
    Posts
    658
    Thanks
    42
    If you fix a point $\displaystyle p$ and express everything in a local chart, then the required path $\displaystyle c(s), \ -\epsilon<s<\epsilon$ is just the unique solution of the initial value problem $\displaystyle c'(0)=D, \ c(0)=p$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2010
    Posts
    131
    Hello Rebesques,

    thank you for your help! But i don't see what you mean. Ok lets choose a chart (U,$\displaystyle \phi$) around the fixed point p.
    I think by your notation you mean $\displaystyle c'(0)=\frac{d}{dt}_{|t=0}(f\circ c)(t)$, is it right?

    But How can i solve this differential equation D=c'(t)?

    But i know by a thm., that every derivation D is of the form
    $\displaystyle D=\sum_{i=1}^d D(x_i) \frac{d}{dx_i}_{|p}$

    whereas $\displaystyle x_i$ are the coordinate functions of our selected chart $\displaystyle \phi$
    Perhaps this equation can help? But i don't know how.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    My house.
    Posts
    658
    Thanks
    42
    Not quite. We are looking for a curve $\displaystyle c-\epsilon,\epsilon)\rightarrow M$ with $\displaystyle c(0)=p, \ c'(0)=(\frac{\partial}{\partial t}\vert_{0})c=D\in T_pM$. There are many curves with this property; Let's construct one.

    Choose a chart $\displaystyle (U,\phi), \ \phi=(x^1,\ldots,x^n)$, and w.l.o.g. assume $\displaystyle \phi(p)=0, \ D(\phi)\vert_{p}=I$, where $\displaystyle D(\phi)\vert_{p}=(\frac{\partial}{\partial x^i}\vert_p(x^j))$ and I is the identity matrix.

    Then, by letting $\displaystyle D=a^i\frac{\partial}{\partial x^i}$ and identifying $\displaystyle D=(a^1,\ldots,a^n)\in R^n$, the curve $\displaystyle c(t)=\phi^{-1}(tD), \ -\epsilon<t<\epsilon$ has the required properties.

    Ps. And $\displaystyle c$ is unique in the sence it belongs to the equivalence class of curves passing through $\displaystyle p$ and having tangent $\displaystyle D\in T_pM$.
    Last edited by Rebesques; Dec 17th 2010 at 02:50 AM. Reason: notation
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Any metric spaces can be viewed as a subset of normed spaces
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Dec 15th 2011, 03:00 PM
  2. tangent spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Apr 21st 2011, 01:21 PM
  3. Tangent Spaces
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Jun 3rd 2010, 07:04 AM
  4. Calculating the dimension of tangent spaces
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Mar 29th 2010, 11:20 PM
  5. Replies: 3
    Last Post: Jun 1st 2008, 01:51 PM

Search Tags


/mathhelpforum @mathhelpforum