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Most treatments of Poisson point processes I have encountered define them on $\mathbb{R}^d$, with the key property that the number of points in a bounded region $R$ will have a Poisson distribution with parameter equal to the integral of the intensity function over the region $R$, independent of the number of points in any bounded region disjoint from $R$.

Can we define a Poisson point process on a Riemannian manifold $(M,g)$? If so, are there any constraints on the kinds of manifolds for which we can define Poisson point processes (orientability, etc)? A text to reference would be very helpful.

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If $(E,\mathcal E,\mu)$ is a non-atomic $\sigma$-finite measure space such that the diagonal in $E\times E$ is $\mathcal E\otimes \mathcal E$-measurable, then there is a Poisson point process on $E$ with intensity measure $\mu$. See section 2.5 in Poisson Processes by J.F.C. Kingman. In particular, a Riemannian manifold supports a Poisson point process with intensity equal to the Riemannian "volume" measure.

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