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I am reading Lorentzian geometry. I found the following definition.

Definition: A Lorentzian manifold $M$ has maximal isometry group if the action of $\text{Isom(M)}$ is transitive and, for every point $x \in M$, every linear isometry $L ∶ T_x M \rightarrow T_xM$ extends to an isometry of $M$.

Now I am confused about what does one mean by extension of linear isometry to an isometry of $M$?

Can anyone please help me?

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"The linear isometry $L:T_{x}M \to T_{x}M$ extends to $M$" means "There exists an isometry $\phi_{L}:M \to M$ such that $D(\phi_{L})(x) = L$."

This is a Lorentzian version of isotropy.

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