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Mathematics
Differential Geometry
Fundamental definition of extrinsic curvature
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[QUOTE="PLuz, post: 6070069, member: 298891"] 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? [/QUOTE]
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Differential Geometry
Fundamental definition of extrinsic curvature
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