Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one left-invariant Riemannian metric such that matrices $$ A_1 = \left( \begin{array}{ccc} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \quad A_2 = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{array} \right) \quad A_3 = \left( \begin{array}{ccc} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array} \right) $$ are the orthonormal basis of $T_I SO(3)$. Can I get some help?
2026-03-26 11:04:30.1774523070
Left-invariant Riemannian metric on $SO(3)$
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HINTS: Consider the function $f(A) = A^\top A$ on the space of $3\times 3$ matrices. Compute $df_I(B)$ and see what it means for this to be $0$.
Next: If you define an orthonormal basis for $T_ISO(3)$, then left-translating it everywhere will give you an orthonormal basis at $T_ASO(3)$ for all $A\in SO(3)$. Does this determine a unique Riemannian metric on $SO(3)$?