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In the paper The geometry of free fall and light propagation by Ehlers and his colleagues (Gen. Relativ. Gravit. 44 no. 6, pp. 1587–1609 (2012)), I reach to an axiom which says:

There exists a collection of triplets $(U,P,P^{\prime})$ where $U\subset M$, $P, P^{\prime}\in \mathscr{P}$ such that the system of maps $x_{PP^\prime}|_U$ is a smooth atlas for $M$.

In the axiom above $\mathscr{P}$ is a subset of $M$ called whose members are world lines of particles, and $x_{PP^\prime}:M\to \mathbb{R}^4$ is a map that sends every $e\in M$ to $(u,v,u^{\prime},v^{\prime})\in\mathbb{R}^4$ which is called radar coordinate.(following figure)

enter image description here

Afterwards, it is concluded according to the axiom that the dimension of the tangent space of $M$ at a point equals the dimension of $M$.


I know the mathematical proof of $\dim T_p M=\dim M$, but I cannot understand how one concludes it from that axiom. Can anyone help me?

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These axioms make M into a diff. manifold. And that is true for every manifold. – MBN Aug 19 '13 at 12:45

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