Suppose $g_1$ and $g_2$ are two metrics defining the same angles, which means $g_1(X,Y)/(g_1(X,X)g_1(Y,Y))^{0.5}=g_2(X,Y)/(g_2(X,X)g_2(Y,Y))^{0.5}$ for all pairs of vector $X,Y$.I want to prove that $g_1=cg_2$ for a constant $c$. I have tried to use the polarization identity，but it doesn't work.
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I think, you meant that $c$ is a function on your manifold rather than constant. Then, it is just a fact from linear algebra: if $A$ is an invertible linear transformation of $R^n$ which preserves angles, then $A$ has the form $cU$, where $c$ is a scalar and $U\in O(n)$. Do you need help with proving the linear algebra statement? – Moishe Kohan Apr 22 '14 at 17:23
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Similar questions have been asked (and answered!) a number of times: here, here and here, to name a few. – Yuri Vyatkin Apr 23 '14 at 08:06
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Vote to close as a duplicate? – Moishe Kohan Apr 23 '14 at 19:52