Let $\{\alpha_0, \ldots, \alpha_r\} \subset \mathbb{Z}^n$ be a finite subset of lattice points and let $\Phi: (\mathbb{C}^*)^n \to \mathbb{C}\mathbb{P}^r$ be the corresponding map from the algebraic torus to the projective space defined by: $$\Phi(x) = (x^{\alpha_0}: \cdots : x^{\alpha_r}).$$ Here $x^\alpha$ is short for $x_1^{a_1} \cdots x_n^{a_n}$, $\alpha = (a_1, \ldots, a_n)$. For simplicity let us assume that $\Phi$ is an embedding.

Consider the standard Fubini-Study metric on $\mathbb{C}\mathbb{P}^r$ and equip the torus $(\mathbb{C}^*)^n$ with the pull-back metric which we call $g$. It is clearly a metric invariant by the action of the compact subgroup $(S^1)^n$.

Let us decompose $(\mathbb{C}^*)^n$ as $\mathbb{R}^n \times (S^1)^n$ by $(r, \theta) \mapsto e^{r+i\theta}$. Then we can consider $\mathbb{R}^n$ as a subset of $(\mathbb{C}^*)^n$.

My question is if one can describe the geodesics of $g$ on $\mathbb{R}^n$? Is it true that the geodesics are straight lines?


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