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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- Dec 20th 2008, 12:29 AMsah_matdifferentiable 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. - Dec 20th 2008, 11:49 PMLaurent
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: $\displaystyle F$ is*continuously*differentiable. Did you forget it?

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

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

In order to get the reverse inequality, prove that, at any point, $\displaystyle F$ has locally an inverse function (by the inverse function theorem), notice that this inverse function satisfies the same hypothesis as $\displaystyle F$ and procede like above to find $\displaystyle \|F^{-1}(f(x))-F^{-1}(f(y))\|\leq \|f(x)-f(y)\|$ if $\displaystyle x,y$ are in a (possibly small) open convex set where $\displaystyle F$ is invertible. And this is it. - Dec 21st 2008, 02:49 AMsah_mat
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.