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Math Help - differentiable map and local isometry

  1. #1
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    differentiable map and local isometry

    hi dear mathhelpform's citizens ı have a question,ı want to ask you;

    prove;
    if the differential of a differentiable map F preserves,orthonormal basis then F is a(local) isometry.
    thanks for your helps.
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  2. #2
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    Quote Originally Posted by sah_mat View Post
    hi dear mathhelpform's citizens ı have a question,ı want to ask you;

    prove;
    if the differential of a differentiable map F preserves,orthonormal basis then F is a(local) isometry.
    thanks for your helps.
    Why did you put parentheses on "local"? It is true with or without, but it is much easier with the parentheses, and you would have been given a hint to do the global version, so I guess what you need is the local version and that's what I'll be writing about.
    I'll need an additional hypothesis: F is continuously differentiable. Did you forget it?


    Note that if \varphi is a linear map, " \varphi preserves orthornormal bases" implies \|\varphi(x)\|=\|x\| for every x, (one says that \varphi is orthogonal). This implies \|\varphi\|=\max_{x\neq 0}\frac{\|\varphi(x)\|}{\|x\|}=1.

    Suppose F is defined on an open convex set U. By the previous remark, we have \|dF_x\|=1 for every x\in U. As a consequence, for any x,y\in U, the mean-value theorem gives \|F(x)-F(y)\|\leq \max_{z\in[x,y]}\|dF_z\|\times \|x-y\|=\|x-y\|.

    In order to get the reverse inequality, prove that, at any point, F has locally an inverse function (by the inverse function theorem), notice that this inverse function satisfies the same hypothesis as F and procede like above to find \|F^{-1}(f(x))-F^{-1}(f(y))\|\leq \|f(x)-f(y)\| if x,y are in a (possibly small) open convex set where F is invertible. And this is it.
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  3. #3
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    thanks laurent,god save you! ı know the isometry of f hold the local isometry but my teacher wants everything in order anyway ı appreciate u ı can handle on this from here thanks again.
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