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Determinantal point processes exhibit an inherent repulsive behavior, thusproviding examples of very evenly distributed point sets on manifolds. In thispaper, we study the so-called harmonic ensemble, defined in terms of Laplaceeigenfunctions on the sphere $\mathbb{S}^d$ and the flat torus $\mathbb{T}^d$,and the so-called spherical ensemble on $\mathbb{S}^2$, which originates inrandom matrix theory. We extend results of Beltr\'an, Marzo and Ortega-Cerd\`aon the Riesz $s$-energy of the harmonic ensemble to the nonsingular regime$s<0$, and as a corollary find the expected value of the spherical cap $L^2$discrepancy via the Stolarsky invariance principle. We also show that thespherical ensemble and the harmonic ensemble on $\mathbb{S}^2$ and$\mathbb{T}^2$ with $N$ points attain the optimal rate $N^{-1/2}$ inexpectation in the Wasserstein metric $W_2$, in contrast to i.i.d. randompoints, which are known to lose a factor of $(\log N)^{1/2}$.

Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus