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Given a pseudo-Riemannian manifold $M$, let say for example Minkowski spacetime in $s>1$ dimensions, often we talk about integration on it. Namely the connection/metric definition gives us a measure on which we integrate. However, to me it is not so clear how to compute integrals on pseudo-Riemannian manifolds in full generality. For example (and these are my main questions): how does the change of variables work? What can we integrate over such a manifold?

A clear and detailed reference on integration on pseudo-Riemannian manifolds is warmly welcome.

Thank you!

HaroldF
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  • Are you familiar with the integration of differential forms on manifolds? The definition of integration on (pseudo-)Riemannian manifolds usually starts there. – Kajelad Aug 06 '20 at 13:53
  • More or less yes, in the orientable case we take a differential form, then choose coordinate patches and a partition of the unity, then we split the integral over the domain as an integral over coordinate patches weighted by the partition function, in this coordinate representation the measure is just the n-dimensional measure $dx_1\cdots dx_n$ weighted by the Radon-Nikodym derivative induced by the differential form. – HaroldF Aug 06 '20 at 13:59
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    Every (pseudo-)Riemannian manifold has a Riemannian density $\mu_g$ which allows one to integrate smooth functions. Chapter 16 (Integration on Manifolds) of Lee's Introduction to Smooth Manifolds, while not terribly self-contained, gives a pretty thorough introduction, assuming some familiarity with manifolds/differential forms . This can in turn be used to define the Riemannian measure, which generalizes the Lebesgue measure on $\mathbb{R}^n$ to Riemannian manifolds. This answer describes the construction of this measure. – Kajelad Aug 06 '20 at 18:55

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