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My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface?

Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an hypersurface orthogonal congruence of time-like curves has zero vorticity and the many definitions that I've seen assume this fact. My guess is that the fundamental definition should be:

$$K_{ab}=\frac{1}{2}\mathcal{L}_n~h_{ab}~,$$

where ##h_{ab}## represents the induced metric on the hypersurface and ##\mathcal{L}_n## the Lie derivative along the normal to the hypersurface. By the way, should there be a minus sign in the above expression, I have seen both cases and it should not be irrelevant?

Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an hypersurface orthogonal congruence of time-like curves has zero vorticity and the many definitions that I've seen assume this fact. My guess is that the fundamental definition should be:

$$K_{ab}=\frac{1}{2}\mathcal{L}_n~h_{ab}~,$$

where ##h_{ab}## represents the induced metric on the hypersurface and ##\mathcal{L}_n## the Lie derivative along the normal to the hypersurface. By the way, should there be a minus sign in the above expression, I have seen both cases and it should not be irrelevant?

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