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Given a submanifold $X$, the second fundamental form tells you about how the submanifold is embedded in the ambient space (intuitively by measuring how a normal vector field varies from point to point.)

It is well known that the trace-free part of the second fundamental form is conformally invariant. That is, if the ambient metric is conformally rescaled, the trace-free part of the second fundamental form in this new geometry will be invariant.

I am wondering what the trace-free part of the second fundamental form corresponds to in general relativity. One can show that if the trace-free part of the second fundamental form does not asymptotically vanish in an asymptotically flat space-time, then the submanifold will not conformally compactify. I believe it should be related to mass in some sense.

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