4
$\begingroup$

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?

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.