Let $M$ be a smooth Riemannian manifold of dimension $d$. I wish to choose in a measurable way a map $C_x:T_xM\rightarrow \mathbb{R}^d$ s.t. $$\forall u,v\in T_xM: \langle C_xu,C_xv\rangle=\langle u,v\rangle_x'$$ where $\langle\cdot , \cdot\rangle_x'$ is some inner-product on the tangent space, which is defined as a function of the Riemannian metric inner-product over some measurable subset of $M$. I wish to choose a measurable $C_x$ in its dependence on $x$, regarding the Borel measure space $M$. I've been struggling and stuck with this for a while, and any hint would be appreciated.
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$\begingroup$ I'm sorry you feel that way. I'd be happy to reformulate it if you tell what specifically bothers you? $\endgroup$– JustSomeGuyCommented Aug 3, 2016 at 14:57
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$\begingroup$ I removed the hyperbolic dynamics tag since I couldn't see any clear connections to dynamical systems (either general or hyperbolic). $\endgroup$– Ian MorrisCommented Aug 3, 2016 at 15:36
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1$\begingroup$ BOS, I started editing your question, but I found myself rewriting it completely and I wouldn't want to do that, so I stopped. Instead I'll try to explain. It is hard to convey a tone. Please trust me, I don't mean to offend. $\endgroup$– Uri BaderCommented Aug 3, 2016 at 19:16
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$\begingroup$ I can read your question in two ways. Either $V$ is a fixed vector space and by $V_x$ you mean $V$ endowed with an inner product which varies measurably according to $x\in M$ (so you have a map $M\to$ inner products on $V$). Or, $V_x$ is a measurable collection of inner product spaces, whatever that means. In the former case you seek a measurable $T:M\to \text{Hom}(V,W)$ and everything is lightly easier to formulate, but non of these cases is really compatible with what you wrote. $\endgroup$– Uri BaderCommented Aug 3, 2016 at 19:27
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$\begingroup$ Ive edited and added details that I found unrelated before. I hope it is clear now. $\endgroup$– JustSomeGuyCommented Aug 4, 2016 at 13:01
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1 Answer
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Use a selection theorem to choose measurably a unit vector in $V_x$. Consider now a new measurable collection of vector spaces $V'_x$ given as the orthogonal complements of the chosen vectors. Iterate $d$-times. By now you got a measurable choice of orthogonal basis. Fix an orthonormal basis in $W$. Use these bases to define $T_x$.
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$\begingroup$ After the current edit I would write my answer as follows: $TM$ is locally trivial, so find a measurable partition $\{U\}$ such that $TM|_U$ is trivial. Enough to work separately on each part, thus we can assume that $TU\simeq U\times \mathbb{R}^d$, and you have a measurable map $U\to$ inner products on $\mathbb{R}^d$. Take the standard basis of $\mathbb{R}^d$ at each $u\in U$ and form a Grahm-Schmidt process. You will obtain an orthonormal basis at each $u$ that varies measurably. Use it to form your linear map. $\endgroup$ Commented Aug 4, 2016 at 13:30