I would appreciate if someone can confirm or correct my understanding of extrinsic-curvature (as in the ADM 3+1 decomposition of spacetime) when dealing with an asymptotically flat spacetime.

I followed mathematical derivations in literature (such as "3+1 Formalism and Bases of Numerical Relativity" ch. 8 by Gourgoulhon (2007)) and I read the answer in this thread.

Is it correct to infer that in an asymptotically-flat spacetime that stretches out to infinity - the extrinsic curvature tensor $K_{mn}$ vanishes (all of its components) at infinity?

I would appreciate if someone can confirm or correct my understanding of extrinsic-curvature (as in the ADM 3+1 decomposition of spacetime) when dealing with a conformally-flat spacetime. (I updated the question's title and body after comments were rightfully made)

I followed mathematical derivations in literature (such as "3+1 Formalism and Bases of Numerical Relativity" ch. 8 by Gourgoulhon (2007)) and I read the answer in this thread. I assume that a conformally-flat spacetime is also asymptotically-flat if the factor function ($\Psi$ usually) is bound. Perhaps this is even trivial to mention, but I'll try to be precise.

Is it correct to infer that in a conformally-flat spacetime that is asymptotically-flat and stretches out to infinity - the extrinsic curvature tensor $K_{mn}$ vanishes (all of its components) at infinity?

I would appreciate if someone can confirm or correct my understanding of extrinsic-curvature (as in the ADM 3+1 decomposition of spacetime) when dealing with an asymptotically flat spacetime.

I followed mathematical derivations in literature (such as "3+1 Formalism and Bases of Numerical Relativity" ch. 8 by Gourgoulhon (2007)) and I read the answer in this thread.

Is it correct to infer that in an asymptotically-flat spacetime that stretches out to infinity - the extrinsic curvature tensor $K_{mn}$ vanishes (all of its components) at infinity?

I would appreciate if someone can confirm or correct my understanding of extrinsic-curvature (as in the ADM 3+1 decomposition of spacetime) when dealing with an asymptotically flat spacetime.

I followed mathematical derivations in literature (such as "3+1 Formalism and Bases of Numerical Relativity" ch. 8 by Gourgoulhon (2007)) and I read the answer in this thread.

Is it correct to infer that in an asymptotically-flat spacetime that stretches out to infinity - the extrinsic curvature tensor $K_{mn}$ vanishes (all of its components) at infinity?