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I am having difficulty understanding something in the book Introduction to Curvature by John M. Lee. Let $\sigma: S^n-N \rightarrow \mathbb{R}^n$ be the stereographic projection, $g_0$ be the metric on $S^n$ induced from the Euclidean metric on $\mathbb{R}^{n+1}$ and $g$ be the Euclidean metric on $\mathbb{R}^n$. Then he proves the conformal equivalance of two metrics by proving that for all $V$ in the tangent space of $\mathbb{R}^n$

$$g_0(\sigma^{-1}_*V,\sigma^{-1}_*V) = f g(V,V)$$

He uses the same method to prove also the equivalence of the Minkowski metric restricted to Hyperboloid and the Poincaré metric on the ball and so on. This confuses me because he always just shows on the same vector field $V$. Isn't he supposed to show it for all pairs $V,U$?

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  • $\begingroup$ Okay I got it from a friend, you use it on $g_0(\sigma^{-1}_*(U+V),\sigma^{-1}_*(U+V)) = f g(U+V,U+V)$ to get $g_0(\sigma^{-1}_*U,\sigma^{-1}_*V) = f g(U,V)$ $\endgroup$ – Sina Mar 10 '15 at 12:18
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Your comment is correct. By the Polarization identity, a symmetric bilinear form is determined by its values on the "diagonal" $u=v$. Hence, to show that two symmetric bilinear forms $g_1,g_2$ on some vector space (such as the tangent space at a point) satisfy $g_1=kg_2$, it suffices to verify $g_1(v,v)=kg_2(v,v)$ for all $v$.

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