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Is there a left-invariant Riemannian metric on $\text{SL}_3(\mathbb{R})$ for which the geodesics (with respect to the corresponding Levi-Civita connection) through the identity are exactly the integral curves for the invariant, smooth vector fields?

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I believe that a standard computation for the inverse problem in the calculus of variations will show that, up to constant multiples, the only pseudo-Riemannian metric on $\mathrm{SL}(3,\mathbb{R})$ whose geodesics are the $1$-parameter subgroups (which is what you are asking) is the usual bi-invariant one, and that one is not positive definite. In particular, this would answer your question in the negative.

For references on the inverse problem in the calculus of variations and a guide to the calculations necessary, I would suggest: Anderson, Ian(1-UTS); Thompson, Gerard(1-TLD), The inverse problem of the calculus of variations for ordinary differential equations. Mem. Amer. Math. Soc. 98 (1992), no. 473, vi+110 pp.

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