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Let $G$ be a compact linear group and $G^c$ be its complexification. Then there is a diffeomorphism $f: G^c \to G \times Lie(G) $ given by $$ x e^{iA} \to (x,A).$$ Let $h$ be the pull back metric of the product metric on $G \times Lie(G)$. Then $h$ has nonnegative sectional curvature. However, $h$ may not be left invariant under $G^c$ since $f$ in general is not a group homomorphism.

My question is: Is there a way to construct a left invariant metric on $G^c$ with nonnegative sectional curvature?

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