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Tangent Space Definition (Munkres Analysis on Manifolds)

  1. Jul 4, 2012 #1
    Hi all,

    I'm quite confused concerning the definition of tangent vectors and tangent spaces as presented in Munkres's Analysis on Manifolds. Here is the book's definition:

    Given ##\textbf{x} \in \mathbb{R}^n##, we define a tangent vector to ##\mathbb{R}^n## at ##\textbf{x}## to be a pair ##(\textbf{x};\textbf{v})##, where ##\textbf{v} \in \mathbb{R}^n##. The set of all tangent vectors to ##\mathbb{R}^n## at ##\textbf{x}## forms a vector space if we define

    ##(\textbf{x}; \textbf{v}) + (\textbf{x};\textbf{w}) = (\textbf{x}; \textbf{v + w})##,
    ##c(\textbf{x};\textbf{v}) = (\textbf{x};c\textbf{v})##.

    It is called the tangent space to ##\mathbb{R}^n## at ##\textbf{x}##, and is denoted ##T_x(\mathbb{R}^n)##.


    I'm not quite sure what is meant by the definition. First off, I'm not sure i understand the notation ##(\textbf{x};\textbf{v})##, which appears to be an ordered pair of vectors. Next, Munkres goes on to describe ##(\textbf{x};\textbf{v})## as "an arrow with its initial point at ##\textbf{x}##, with ##T_x(\mathbb{R}^n)## as the set of all arrows with their initial point at ##\textbf{x}##. What I don't understand is from this description, isn't ##T_x(\mathbb{R}^n## just spanning all of ##\mathbb{R}^n)##? What is the distinction between the two?

    Munkres also describes the set ##T_x(\mathbb{R}^n)## as "just the set ##\textbf{x} \times \mathbb{R}^n##. I am also unfamiliar with this notation. If ##\textbf{x}## is also in ##\mathbb{R}^n##, then is ##T_x(\mathbb{R}^n)## a subset of ##\mathbb{R}^{2n}##? That doesn't seem right to me, although I just don't know how to interpret Munkres's explanation.

    I think my problem lies in my not understanding the notation used in this chapter. Perhaps the best way for me to understand is maybe with an example. If ##\textbf{x} = (1,1) \in \mathbb{R}^2##, then what is ##T_x(\mathbb{R}^2)## according to Munkres's definition?

    Thanks so much for the help all!
     
  2. jcsd
  3. Jul 4, 2012 #2

    tiny-tim

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    hi mathmonkey! :smile:
    i think it's easier if instead of ℝn, we use the surface, S, of an egg

    then, for each point x on S, TxS (or Tx(S)) is the tangent plane at x

    although S is round, TxS is flat, and is isomorphic to ℝ2

    TxS and TyS are isomorphic to each other, even if x and y are at points of the egg of different curvature

    S has nothing to do with ℝ2 (except that they're there same dimension)

    Mukres's example has ℝ2 instead of S, so in that particular case the manifold and the tangent space are isomorphic, but they still have nothing to do with each other :wink:
    he's just saying that each tangent plane is defined not just as the ℝ2 that it's isomorphic to, but also by the point of attachment, x
     
  4. Jul 4, 2012 #3
    Hi tiny-tim,

    Thanks for your help. So am I right in understanding that in my above example where ##x = (1,1) \in \mathbb{R}^2##, then ##T_x(\mathbb{R}^2)## is in fact the entire plane ##\mathbb{R}^2##, where each of the pairs ##(\textbf{x};\textbf{v})## is a vector ##\textbf{v}## originating from the point ##\textbf{x}##? I have not yet gotten to the part of the text relating this definition to manifolds, so perhaps right now this definition seems to me for the time being unmotivated, especially as I already have preconceived notions of what a "tangent" space looks like based on its name. But I just wanted to make sure I understood the formal definition clearly.

    Thanks again for your help!
     
  5. Jul 4, 2012 #4

    tiny-tim

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    yes, it's an entire plane :smile:

    (you shouldn't think of all the tangent planes as being the same … they're not the same thing, they're identical different things :wink:)
     
  6. Jul 4, 2012 #5
    Great, thanks for your help! I just wanted to make sure I understood fully before moving on in the chapter!
     
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