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Can someone please explain the difference between local rigidity and infinitesimal rigidity? Does either version of rigidity imply the other?

In particular, I'm thinking about Weil's rigidity theorem for hyperbolic metrics on manifolds of dimension $\geq 3$. I've seen it referred to as both local and infinitesimal, which further adds to my confusion about the distinction.

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# Local vs. infinitesimal rigidity

Can someone please explain the difference between local rigidity and infinitesimal rigidity? In particular, I'm thinking about Weil's rigidity theorem for hyperbolic metrics on manifolds of dimension $\geq 3$. I've seen it referred to as both local and infinitesimal, which further adds to my confusion about the distinction.
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